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Mirrors > Home > MPE Home > Th. List > funopabeq | Structured version Visualization version GIF version |
Description: A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.) |
Ref | Expression |
---|---|
funopabeq | ⊢ Fun {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funopab 6383 | . 2 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝐴} ↔ ∀𝑥∃*𝑦 𝑦 = 𝐴) | |
2 | moeq 3695 | . 2 ⊢ ∃*𝑦 𝑦 = 𝐴 | |
3 | 1, 2 | mpgbir 1791 | 1 ⊢ Fun {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∃*wmo 2613 {copab 5119 Fun wfun 6342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-fun 6350 |
This theorem is referenced by: funopab4 6385 |
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