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Theorem dom2lem 6188
Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.)
Hypotheses
Ref Expression
dom2d.1 (φ → (x A𝐶 B))
dom2d.2 (φ → ((x A y A) → (𝐶 = 𝐷x = y)))
Assertion
Ref Expression
dom2lem (φ → (x A𝐶):A1-1B)
Distinct variable groups:   x,y,A   x,B,y   y,𝐶   x,𝐷   φ,x,y
Allowed substitution hints:   𝐶(x)   𝐷(y)

Proof of Theorem dom2lem
StepHypRef Expression
1 dom2d.1 . . . 4 (φ → (x A𝐶 B))
21ralrimiv 2385 . . 3 (φx A 𝐶 B)
3 eqid 2037 . . . 4 (x A𝐶) = (x A𝐶)
43fmpt 5262 . . 3 (x A 𝐶 B ↔ (x A𝐶):AB)
52, 4sylib 127 . 2 (φ → (x A𝐶):AB)
61imp 115 . . . . . . 7 ((φ x A) → 𝐶 B)
73fvmpt2 5197 . . . . . . . 8 ((x A 𝐶 B) → ((x A𝐶)‘x) = 𝐶)
87adantll 445 . . . . . . 7 (((φ x A) 𝐶 B) → ((x A𝐶)‘x) = 𝐶)
96, 8mpdan 398 . . . . . 6 ((φ x A) → ((x A𝐶)‘x) = 𝐶)
109adantrr 448 . . . . 5 ((φ (x A y A)) → ((x A𝐶)‘x) = 𝐶)
11 nfv 1418 . . . . . . . 8 x(φ y A)
12 nffvmpt1 5129 . . . . . . . . 9 x((x A𝐶)‘y)
1312nfeq1 2184 . . . . . . . 8 x((x A𝐶)‘y) = 𝐷
1411, 13nfim 1461 . . . . . . 7 x((φ y A) → ((x A𝐶)‘y) = 𝐷)
15 eleq1 2097 . . . . . . . . . 10 (x = y → (x Ay A))
1615anbi2d 437 . . . . . . . . 9 (x = y → ((φ x A) ↔ (φ y A)))
1716imbi1d 220 . . . . . . . 8 (x = y → (((φ x A) → ((x A𝐶)‘x) = 𝐶) ↔ ((φ y A) → ((x A𝐶)‘x) = 𝐶)))
1815anbi1d 438 . . . . . . . . . . . 12 (x = y → ((x A y A) ↔ (y A y A)))
19 anidm 376 . . . . . . . . . . . 12 ((y A y A) ↔ y A)
2018, 19syl6bb 185 . . . . . . . . . . 11 (x = y → ((x A y A) ↔ y A))
2120anbi2d 437 . . . . . . . . . 10 (x = y → ((φ (x A y A)) ↔ (φ y A)))
22 fveq2 5121 . . . . . . . . . . . . 13 (x = y → ((x A𝐶)‘x) = ((x A𝐶)‘y))
2322adantr 261 . . . . . . . . . . . 12 ((x = y (φ (x A y A))) → ((x A𝐶)‘x) = ((x A𝐶)‘y))
24 dom2d.2 . . . . . . . . . . . . . 14 (φ → ((x A y A) → (𝐶 = 𝐷x = y)))
2524imp 115 . . . . . . . . . . . . 13 ((φ (x A y A)) → (𝐶 = 𝐷x = y))
2625biimparc 283 . . . . . . . . . . . 12 ((x = y (φ (x A y A))) → 𝐶 = 𝐷)
2723, 26eqeq12d 2051 . . . . . . . . . . 11 ((x = y (φ (x A y A))) → (((x A𝐶)‘x) = 𝐶 ↔ ((x A𝐶)‘y) = 𝐷))
2827ex 108 . . . . . . . . . 10 (x = y → ((φ (x A y A)) → (((x A𝐶)‘x) = 𝐶 ↔ ((x A𝐶)‘y) = 𝐷)))
2921, 28sylbird 159 . . . . . . . . 9 (x = y → ((φ y A) → (((x A𝐶)‘x) = 𝐶 ↔ ((x A𝐶)‘y) = 𝐷)))
3029pm5.74d 171 . . . . . . . 8 (x = y → (((φ y A) → ((x A𝐶)‘x) = 𝐶) ↔ ((φ y A) → ((x A𝐶)‘y) = 𝐷)))
3117, 30bitrd 177 . . . . . . 7 (x = y → (((φ x A) → ((x A𝐶)‘x) = 𝐶) ↔ ((φ y A) → ((x A𝐶)‘y) = 𝐷)))
3214, 31, 9chvar 1637 . . . . . 6 ((φ y A) → ((x A𝐶)‘y) = 𝐷)
3332adantrl 447 . . . . 5 ((φ (x A y A)) → ((x A𝐶)‘y) = 𝐷)
3410, 33eqeq12d 2051 . . . 4 ((φ (x A y A)) → (((x A𝐶)‘x) = ((x A𝐶)‘y) ↔ 𝐶 = 𝐷))
3525biimpd 132 . . . 4 ((φ (x A y A)) → (𝐶 = 𝐷x = y))
3634, 35sylbid 139 . . 3 ((φ (x A y A)) → (((x A𝐶)‘x) = ((x A𝐶)‘y) → x = y))
3736ralrimivva 2395 . 2 (φx A y A (((x A𝐶)‘x) = ((x A𝐶)‘y) → x = y))
38 nfmpt1 3841 . . 3 x(x A𝐶)
39 nfcv 2175 . . 3 y(x A𝐶)
4038, 39dff13f 5352 . 2 ((x A𝐶):A1-1B ↔ ((x A𝐶):AB x A y A (((x A𝐶)‘x) = ((x A𝐶)‘y) → x = y)))
415, 37, 40sylanbrc 394 1 (φ → (x A𝐶):A1-1B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  wral 2300  cmpt 3809  wf 4841  1-1wf1 4842  cfv 4845
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-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
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-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  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-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-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fv 4853
This theorem is referenced by:  dom2d  6189  dom3d  6190
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