f1dmex - Metamath Proof Explorer

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Theorem f1dmex 7657

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 5189. (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 6574	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2	1	frnd 6520	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵)
3		ssexg 5226	. . . . 5 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → ran 𝐹 ∈ V)
4	2, 3	sylan 582	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝐶) → ran 𝐹 ∈ V)
5	4	ex 415	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐵 ∈ 𝐶 → ran 𝐹 ∈ V))
6		f1cnv 6637	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → ^◡𝐹:ran 𝐹–1-1-onto→𝐴)
7		f1ofo 6621	. . . 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		fornex 7656	. . 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 409	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 ^◡ccnv 5553 ran crn 5555 –1-1→wf1 6351 –onto→wfo 6352 –1-1-onto→wf1o 6353

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pr 5329 ax-un 7460

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362

This theorem is referenced by: f1ovv 7658 f1domg 8528 ordtypelem10 8990 oiexg 8998 inf3lem7 9096 pwfseqlem4 10083 pwfseqlem5 10084 grothomex 10250 gsumzf1o 19031 dprdf1o 19153 f1lindf 20965 tsmsf1o 22752 diophrw 39354

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