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Theorem elpmi 8776

Description: A partial function is a function. (Contributed by Mario Carneiro, 15-Sep-2015.)

**Assertion**
Ref	Expression
elpmi	⊢ (𝐹 ∈ (𝐴 ↑_pm 𝐵) → (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵))

**Proof of Theorem elpmi**
Step	Hyp	Ref	Expression
1		n0i 4289	. . . 4 ⊢ (𝐹 ∈ (𝐴 ↑_pm 𝐵) → ¬ (𝐴 ↑_pm 𝐵) = ∅)
2		fnpm 8764	. . . . . 6 ⊢ ↑_pm Fn (V × V)
3	2	fndmi 6591	. . . . 5 ⊢ dom ↑_pm = (V × V)
4	3	ndmov 7536	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ↑_pm 𝐵) = ∅)
5	1, 4	nsyl2 141	. . 3 ⊢ (𝐹 ∈ (𝐴 ↑_pm 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6		elpm2g 8774	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹 ∈ (𝐴 ↑_pm 𝐵) ↔ (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵)))
7	5, 6	syl 17	. 2 ⊢ (𝐹 ∈ (𝐴 ↑_pm 𝐵) → (𝐹 ∈ (𝐴 ↑_pm 𝐵) ↔ (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵)))
8	7	ibi 267	1 ⊢ (𝐹 ∈ (𝐴 ↑_pm 𝐵) → (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 Vcvv 3436 ⊆ wss 3897 ∅c0 4282 × cxp 5617 dom cdm 5619 ⟶wf 6483 (class class class)co 7352 ↑_pm cpm 8757

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-fv 6495 df-ov 7355 df-oprab 7356 df-mpo 7357 df-1st 7927 df-2nd 7928 df-pm 8759

This theorem is referenced by: pmfun 8777 pmresg 8800 equivcau 25233 dvn2bss 25865 mrsubff 35563 mrsubrn 35564 elpmrn 45322 elpmi2 45327 issmflem 46830

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