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Theorem dom3d 6190
 Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by Mario Carneiro, 20-May-2013.)
Hypotheses
Ref Expression
dom2d.1 (φ → (x A𝐶 B))
dom2d.2 (φ → ((x A y A) → (𝐶 = 𝐷x = y)))
dom3d.3 (φA 𝑉)
dom3d.4 (φB 𝑊)
Assertion
Ref Expression
dom3d (φAB)
Distinct variable groups:   x,y,A   x,B,y   y,𝐶   x,𝐷   φ,x,y
Allowed substitution hints:   𝐶(x)   𝐷(y)   𝑉(x,y)   𝑊(x,y)

Proof of Theorem dom3d
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 dom2d.1 . . . . . 6 (φ → (x A𝐶 B))
2 dom2d.2 . . . . . 6 (φ → ((x A y A) → (𝐶 = 𝐷x = y)))
31, 2dom2lem 6188 . . . . 5 (φ → (x A𝐶):A1-1B)
4 f1f 5035 . . . . 5 ((x A𝐶):A1-1B → (x A𝐶):AB)
53, 4syl 14 . . . 4 (φ → (x A𝐶):AB)
6 dom3d.3 . . . 4 (φA 𝑉)
7 dom3d.4 . . . 4 (φB 𝑊)
8 fex2 5002 . . . 4 (((x A𝐶):AB A 𝑉 B 𝑊) → (x A𝐶) V)
95, 6, 7, 8syl3anc 1134 . . 3 (φ → (x A𝐶) V)
10 f1eq1 5030 . . . 4 (z = (x A𝐶) → (z:A1-1B ↔ (x A𝐶):A1-1B))
1110spcegv 2635 . . 3 ((x A𝐶) V → ((x A𝐶):A1-1Bz z:A1-1B))
129, 3, 11sylc 56 . 2 (φz z:A1-1B)
13 brdomg 6165 . . 3 (B 𝑊 → (ABz z:A1-1B))
147, 13syl 14 . 2 (φ → (ABz z:A1-1B))
1512, 14mpbird 156 1 (φAB)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551   class class class wbr 3755   ↦ cmpt 3809  ⟶wf 4841  –1-1→wf1 4842   ≼ cdom 6156 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-bndl 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-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  df-dom 6159 This theorem is referenced by:  dom3  6192  xpdom2  6241  fopwdom  6246
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