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Theorem f1dmex 7398
Description: If the codomain of a one-to-one function exists, so does its domain. This theorem is equivalent to the Axiom of Replacement ax-rep 4994. (Contributed by NM, 4-Sep-2004.)
Assertion
Ref Expression
f1dmex ((𝐹:𝐴1-1𝐵𝐵𝐶) → 𝐴 ∈ V)

Proof of Theorem f1dmex
StepHypRef Expression
1 f1f 6338 . . . . . 6 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
21frnd 6285 . . . . 5 (𝐹:𝐴1-1𝐵 → ran 𝐹𝐵)
3 ssexg 5029 . . . . 5 ((ran 𝐹𝐵𝐵𝐶) → ran 𝐹 ∈ V)
42, 3sylan 575 . . . 4 ((𝐹:𝐴1-1𝐵𝐵𝐶) → ran 𝐹 ∈ V)
54ex 403 . . 3 (𝐹:𝐴1-1𝐵 → (𝐵𝐶 → ran 𝐹 ∈ V))
6 f1cnv 6401 . . . 4 (𝐹:𝐴1-1𝐵𝐹:ran 𝐹1-1-onto𝐴)
7 f1ofo 6385 . . . 4 (𝐹:ran 𝐹1-1-onto𝐴𝐹:ran 𝐹onto𝐴)
86, 7syl 17 . . 3 (𝐹:𝐴1-1𝐵𝐹:ran 𝐹onto𝐴)
9 fornex 7397 . . 3 (ran 𝐹 ∈ V → (𝐹:ran 𝐹onto𝐴𝐴 ∈ V))
105, 8, 9syl6ci 71 . 2 (𝐹:𝐴1-1𝐵 → (𝐵𝐶𝐴 ∈ V))
1110imp 397 1 ((𝐹:𝐴1-1𝐵𝐵𝐶) → 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wcel 2164  Vcvv 3414  wss 3798  ccnv 5341  ran crn 5343  1-1wf1 6120  ontowfo 6121  1-1-ontowf1o 6122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131
This theorem is referenced by:  f1ovv  7399  f1domg  8242  ordtypelem10  8701  oiexg  8709  inf3lem7  8808  pwfseqlem4  9799  pwfseqlem5  9800  grothomex  9966  gsumzf1o  18666  dprdf1o  18785  f1lindf  20528  tsmsf1o  22318  diophrw  38159
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