elpmi - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > elpmi		Structured version Visualization version GIF version

Theorem elpmi 8796

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 4299	. . . 4 ⊢ (𝐹 ∈ (𝐴 ↑_pm 𝐵) → ¬ (𝐴 ↑_pm 𝐵) = ∅)
2		fnpm 8784	. . . . . 6 ⊢ ↑_pm Fn (V × V)
3	2	fndmi 6604	. . . . 5 ⊢ dom ↑_pm = (V × V)
4	3	ndmov 7553	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ↑_pm 𝐵) = ∅)
5	1, 4	nsyl2 141	. . 3 ⊢ (𝐹 ∈ (𝐴 ↑_pm 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6		elpm2g 8794	. . 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 1540 ∈ wcel 2109 Vcvv 3444 ⊆ wss 3911 ∅c0 4292 × cxp 5629 dom cdm 5631 ⟶wf 6495 (class class class)co 7369 ↑_pm cpm 8777

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-fv 6507 df-ov 7372 df-oprab 7373 df-mpo 7374 df-1st 7947 df-2nd 7948 df-pm 8779

This theorem is referenced by: pmfun 8797 pmresg 8820 equivcau 25233 dvn2bss 25865 mrsubff 35492 mrsubrn 35493 elpmrn 45207 elpmi2 45212 issmflem 46718

W3C validator