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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 5139 is a function. (Contributed by NM, 17-Feb-2013.) |
Ref | Expression |
---|---|
funopab4 | ⊢ Fun {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝑦 = 𝐴)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴) | |
2 | 1 | ssopab2i 5429 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ (𝜑 ∧ 𝑦 = 𝐴)} ⊆ {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝐴} |
3 | funopabeq 6385 | . 2 ⊢ Fun {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝐴} | |
4 | funss 6368 | . 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 398 = wceq 1533 ⊆ wss 3935 {copab 5120 Fun wfun 6343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-fun 6351 |
This theorem is referenced by: funmpt 6387 hartogslem1 9000 |
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