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Theorem elpmi 8864

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 4334	. . . 4 ⊢ (𝐹 ∈ (𝐴 ↑_pm 𝐵) → ¬ (𝐴 ↑_pm 𝐵) = ∅)
2		fnpm 8852	. . . . . 6 ⊢ ↑_pm Fn (V × V)
3	2	fndmi 6658	. . . . 5 ⊢ dom ↑_pm = (V × V)
4	3	ndmov 7605	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ↑_pm 𝐵) = ∅)
5	1, 4	nsyl2 141	. . 3 ⊢ (𝐹 ∈ (𝐴 ↑_pm 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6		elpm2g 8862	. . 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 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ⊆ wss 3947 ∅c0 4323 × cxp 5676 dom cdm 5678 ⟶wf 6544 (class class class)co 7420 ↑_pm cpm 8845

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-pm 8847

This theorem is referenced by: pmfun 8865 pmresg 8888 equivcau 25227 dvn2bss 25859 mrsubff 35122 mrsubrn 35123 elpmrn 44593 elpmi2 44598 issmflem 46115

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