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Theorem elpmi 8634

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 4267	. . . 4 ⊢ (𝐹 ∈ (𝐴 ↑_pm 𝐵) → ¬ (𝐴 ↑_pm 𝐵) = ∅)
2		fnpm 8623	. . . . . 6 ⊢ ↑_pm Fn (V × V)
3	2	fndmi 6537	. . . . 5 ⊢ dom ↑_pm = (V × V)
4	3	ndmov 7456	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ↑_pm 𝐵) = ∅)
5	1, 4	nsyl2 141	. . 3 ⊢ (𝐹 ∈ (𝐴 ↑_pm 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6		elpm2g 8632	. . 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 266	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 396 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ⊆ wss 3887 ∅c0 4256 × cxp 5587 dom cdm 5589 ⟶wf 6429 (class class class)co 7275 ↑_pm cpm 8616

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-pm 8618

This theorem is referenced by: pmfun 8635 pmresg 8658 equivcau 24464 dvn2bss 25094 mrsubff 33474 mrsubrn 33475 elpmrn 42760 elpmi2 42764 issmflem 44263

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