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Theorem elpmi 8784

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 4281	. . . 4 ⊢ (𝐹 ∈ (𝐴 ↑_pm 𝐵) → ¬ (𝐴 ↑_pm 𝐵) = ∅)
2		fnpm 8772	. . . . . 6 ⊢ ↑_pm Fn (V × V)
3	2	fndmi 6594	. . . . 5 ⊢ dom ↑_pm = (V × V)
4	3	ndmov 7542	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ↑_pm 𝐵) = ∅)
5	1, 4	nsyl2 141	. . 3 ⊢ (𝐹 ∈ (𝐴 ↑_pm 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6		elpm2g 8782	. . 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 1542 ∈ wcel 2114 Vcvv 3430 ⊆ wss 3890 ∅c0 4274 × cxp 5620 dom cdm 5622 ⟶wf 6486 (class class class)co 7358 ↑_pm cpm 8765

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 2709 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-fv 6498 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-pm 8767

This theorem is referenced by: pmfun 8785 pmresg 8809 equivcau 25245 dvn2bss 25875 mrsubff 35700 mrsubrn 35701 elpmrn 45652 elpmi2 45657 issmflem 47159

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