f1dmex - Metamath Proof Explorer

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Theorem f1dmex 7910

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 5212. (Contributed by NM, 4-Sep-2004.)

**Assertion**
Ref	Expression
f1dmex	⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)

**Proof of Theorem f1dmex**
Step	Hyp	Ref	Expression
1		f1f 6736	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2	1	frnd 6676	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵)
3		ssexg 5264	. . . . 5 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → ran 𝐹 ∈ V)
4	2, 3	sylan 581	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝐶) → ran 𝐹 ∈ V)
5	4	ex 412	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐵 ∈ 𝐶 → ran 𝐹 ∈ V))
6		f1cnv 6804	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → ^◡𝐹:ran 𝐹–1-1-onto→𝐴)
7		f1ofo 6787	. . . 4 ⊢ (^◡𝐹:ran 𝐹–1-1-onto→𝐴 → ^◡𝐹:ran 𝐹–onto→𝐴)
8	6, 7	syl 17	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → ^◡𝐹:ran 𝐹–onto→𝐴)
9		focdmex 7909	. . 3 ⊢ (ran 𝐹 ∈ V → (^◡𝐹:ran 𝐹–onto→𝐴 → 𝐴 ∈ V))
10	5, 8, 9	syl6ci 71	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐵 ∈ 𝐶 → 𝐴 ∈ V))
11	10	imp 406	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2114 Vcvv 3429 ⊆ wss 3889 ^◡ccnv 5630 ran crn 5632 –1-1→wf1 6495 –onto→wfo 6496 –1-1-onto→wf1o 6497

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pr 5375 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506

This theorem is referenced by: f1ovv 7911 f1domg 8918 ordtypelem10 9442 oiexg 9450 inf3lem7 9555 pwfseqlem4 10585 pwfseqlem5 10586 grothomex 10752 gsumzf1o 19887 dprdf1o 20009 f1lindf 21802 tsmsf1o 24110 onvf1od 35289 mh-inf3sn 36724 diophrw 43191

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