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Theorem fundmen 6222
 Description: A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 28-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
Hypothesis
Ref Expression
fundmen.1 𝐹 V
Assertion
Ref Expression
fundmen (Fun 𝐹 → dom 𝐹𝐹)

Proof of Theorem fundmen
Dummy variables x y z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fundmen.1 . . . 4 𝐹 V
21dmex 4541 . . 3 dom 𝐹 V
32a1i 9 . 2 (Fun 𝐹 → dom 𝐹 V)
41a1i 9 . 2 (Fun 𝐹𝐹 V)
5 funfvop 5222 . . 3 ((Fun 𝐹 x dom 𝐹) → ⟨x, (𝐹x)⟩ 𝐹)
65ex 108 . 2 (Fun 𝐹 → (x dom 𝐹 → ⟨x, (𝐹x)⟩ 𝐹))
7 funrel 4862 . . 3 (Fun 𝐹 → Rel 𝐹)
8 elreldm 4503 . . . 4 ((Rel 𝐹 y 𝐹) → y dom 𝐹)
98ex 108 . . 3 (Rel 𝐹 → (y 𝐹 y dom 𝐹))
107, 9syl 14 . 2 (Fun 𝐹 → (y 𝐹 y dom 𝐹))
11 df-rel 4295 . . . . . . . . 9 (Rel 𝐹𝐹 ⊆ (V × V))
127, 11sylib 127 . . . . . . . 8 (Fun 𝐹𝐹 ⊆ (V × V))
1312sselda 2939 . . . . . . 7 ((Fun 𝐹 y 𝐹) → y (V × V))
14 elvv 4345 . . . . . . 7 (y (V × V) ↔ zw y = ⟨z, w⟩)
1513, 14sylib 127 . . . . . 6 ((Fun 𝐹 y 𝐹) → zw y = ⟨z, w⟩)
16 inteq 3609 . . . . . . . . . . . . . . . . 17 (y = ⟨z, w⟩ → y = z, w⟩)
1716inteqd 3611 . . . . . . . . . . . . . . . 16 (y = ⟨z, w⟩ → y = z, w⟩)
18 vex 2554 . . . . . . . . . . . . . . . . 17 z V
19 vex 2554 . . . . . . . . . . . . . . . . 17 w V
2018, 19op1stb 4175 . . . . . . . . . . . . . . . 16 z, w⟩ = z
2117, 20syl6eq 2085 . . . . . . . . . . . . . . 15 (y = ⟨z, w⟩ → y = z)
22 eqeq1 2043 . . . . . . . . . . . . . . 15 (x = y → (x = z y = z))
2321, 22syl5ibr 145 . . . . . . . . . . . . . 14 (x = y → (y = ⟨z, w⟩ → x = z))
24 opeq1 3540 . . . . . . . . . . . . . 14 (x = z → ⟨x, w⟩ = ⟨z, w⟩)
2523, 24syl6 29 . . . . . . . . . . . . 13 (x = y → (y = ⟨z, w⟩ → ⟨x, w⟩ = ⟨z, w⟩))
2625imp 115 . . . . . . . . . . . 12 ((x = y y = ⟨z, w⟩) → ⟨x, w⟩ = ⟨z, w⟩)
27 eqeq2 2046 . . . . . . . . . . . . . 14 (⟨x, w⟩ = ⟨z, w⟩ → (y = ⟨x, w⟩ ↔ y = ⟨z, w⟩))
2827biimprcd 149 . . . . . . . . . . . . 13 (y = ⟨z, w⟩ → (⟨x, w⟩ = ⟨z, w⟩ → y = ⟨x, w⟩))
2928adantl 262 . . . . . . . . . . . 12 ((x = y y = ⟨z, w⟩) → (⟨x, w⟩ = ⟨z, w⟩ → y = ⟨x, w⟩))
3026, 29mpd 13 . . . . . . . . . . 11 ((x = y y = ⟨z, w⟩) → y = ⟨x, w⟩)
3130ancoms 255 . . . . . . . . . 10 ((y = ⟨z, w x = y) → y = ⟨x, w⟩)
3231adantl 262 . . . . . . . . 9 (((Fun 𝐹 y 𝐹) (y = ⟨z, w x = y)) → y = ⟨x, w⟩)
3330eleq1d 2103 . . . . . . . . . . . . . . 15 ((x = y y = ⟨z, w⟩) → (y 𝐹 ↔ ⟨x, w 𝐹))
3433adantl 262 . . . . . . . . . . . . . 14 ((Fun 𝐹 (x = y y = ⟨z, w⟩)) → (y 𝐹 ↔ ⟨x, w 𝐹))
35 funopfv 5156 . . . . . . . . . . . . . . 15 (Fun 𝐹 → (⟨x, w 𝐹 → (𝐹x) = w))
3635adantr 261 . . . . . . . . . . . . . 14 ((Fun 𝐹 (x = y y = ⟨z, w⟩)) → (⟨x, w 𝐹 → (𝐹x) = w))
3734, 36sylbid 139 . . . . . . . . . . . . 13 ((Fun 𝐹 (x = y y = ⟨z, w⟩)) → (y 𝐹 → (𝐹x) = w))
3837exp32 347 . . . . . . . . . . . 12 (Fun 𝐹 → (x = y → (y = ⟨z, w⟩ → (y 𝐹 → (𝐹x) = w))))
3938com24 81 . . . . . . . . . . 11 (Fun 𝐹 → (y 𝐹 → (y = ⟨z, w⟩ → (x = y → (𝐹x) = w))))
4039imp43 337 . . . . . . . . . 10 (((Fun 𝐹 y 𝐹) (y = ⟨z, w x = y)) → (𝐹x) = w)
4140opeq2d 3547 . . . . . . . . 9 (((Fun 𝐹 y 𝐹) (y = ⟨z, w x = y)) → ⟨x, (𝐹x)⟩ = ⟨x, w⟩)
4232, 41eqtr4d 2072 . . . . . . . 8 (((Fun 𝐹 y 𝐹) (y = ⟨z, w x = y)) → y = ⟨x, (𝐹x)⟩)
4342exp32 347 . . . . . . 7 ((Fun 𝐹 y 𝐹) → (y = ⟨z, w⟩ → (x = yy = ⟨x, (𝐹x)⟩)))
4443exlimdvv 1774 . . . . . 6 ((Fun 𝐹 y 𝐹) → (zw y = ⟨z, w⟩ → (x = yy = ⟨x, (𝐹x)⟩)))
4515, 44mpd 13 . . . . 5 ((Fun 𝐹 y 𝐹) → (x = yy = ⟨x, (𝐹x)⟩))
4645adantrl 447 . . . 4 ((Fun 𝐹 (x dom 𝐹 y 𝐹)) → (x = yy = ⟨x, (𝐹x)⟩))
47 inteq 3609 . . . . . . . . 9 (y = ⟨x, (𝐹x)⟩ → y = x, (𝐹x)⟩)
4847inteqd 3611 . . . . . . . 8 (y = ⟨x, (𝐹x)⟩ → y = x, (𝐹x)⟩)
4948adantl 262 . . . . . . 7 (((Fun 𝐹 x dom 𝐹) y = ⟨x, (𝐹x)⟩) → y = x, (𝐹x)⟩)
50 vex 2554 . . . . . . . . 9 x V
51 funfvex 5135 . . . . . . . . 9 ((Fun 𝐹 x dom 𝐹) → (𝐹x) V)
52 op1stbg 4176 . . . . . . . . 9 ((x V (𝐹x) V) → x, (𝐹x)⟩ = x)
5350, 51, 52sylancr 393 . . . . . . . 8 ((Fun 𝐹 x dom 𝐹) → x, (𝐹x)⟩ = x)
5453adantr 261 . . . . . . 7 (((Fun 𝐹 x dom 𝐹) y = ⟨x, (𝐹x)⟩) → x, (𝐹x)⟩ = x)
5549, 54eqtr2d 2070 . . . . . 6 (((Fun 𝐹 x dom 𝐹) y = ⟨x, (𝐹x)⟩) → x = y)
5655ex 108 . . . . 5 ((Fun 𝐹 x dom 𝐹) → (y = ⟨x, (𝐹x)⟩ → x = y))
5756adantrr 448 . . . 4 ((Fun 𝐹 (x dom 𝐹 y 𝐹)) → (y = ⟨x, (𝐹x)⟩ → x = y))
5846, 57impbid 120 . . 3 ((Fun 𝐹 (x dom 𝐹 y 𝐹)) → (x = yy = ⟨x, (𝐹x)⟩))
5958ex 108 . 2 (Fun 𝐹 → ((x dom 𝐹 y 𝐹) → (x = yy = ⟨x, (𝐹x)⟩)))
603, 4, 6, 10, 59en3d 6185 1 (Fun 𝐹 → dom 𝐹𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551   ⊆ wss 2911  ⟨cop 3370  ∩ cint 3606   class class class wbr 3755   × cxp 4286  dom cdm 4288  Rel wrel 4293  Fun wfun 4839  ‘cfv 4845   ≈ cen 6155 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-en 6158 This theorem is referenced by:  fundmeng  6223
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