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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 5923 | . 2 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝐴} ↔ ∀𝑥∃*𝑦 𝑦 = 𝐴) | |
2 | moeq 3382 | . 2 ⊢ ∃*𝑦 𝑦 = 𝐴 | |
3 | 1, 2 | mpgbir 1726 | 1 ⊢ Fun {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∃*wmo 2471 {copab 4712 Fun wfun 5882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-fun 5890 |
This theorem is referenced by: funopab4 5925 |
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