![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > funopab4 | Structured version Visualization version GIF version |
Description: A class of ordered pairs of values in the form used by df-mpt 5190 is a function. (Contributed by NM, 17-Feb-2013.) |
Ref | Expression |
---|---|
funopab4 | ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐴)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 486 | . . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴) | |
2 | 1 | ssopab2i 5508 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐴)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴} |
3 | funopabeq 6538 | . 2 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴} | |
4 | funss 6521 | . 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐴)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴} → (Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴} → Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐴)})) | |
5 | 2, 3, 4 | mp2 9 | 1 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐴)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ⊆ wss 3911 {copab 5168 Fun wfun 6491 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-fun 6499 |
This theorem is referenced by: funmpt 6540 hartogslem1 9483 |
Copyright terms: Public domain | W3C validator |