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Mirrors > Home > MPE Home > Th. List > funoprabg | Structured version Visualization version GIF version |
Description: "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 28-Aug-2007.) |
Ref | Expression |
---|---|
funoprabg | ⊢ (∀𝑥∀𝑦∃*𝑧𝜑 → Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mosubopt 5391 | . . 3 ⊢ (∀𝑥∀𝑦∃*𝑧𝜑 → ∃*𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
2 | 1 | alrimiv 1919 | . 2 ⊢ (∀𝑥∀𝑦∃*𝑧𝜑 → ∀𝑤∃*𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
3 | dfoprab2 7201 | . . . 4 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
4 | 3 | funeqi 6369 | . . 3 ⊢ (Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ Fun {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)}) |
5 | funopab 6383 | . . 3 ⊢ (Fun {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} ↔ ∀𝑤∃*𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
6 | 4, 5 | bitr2i 277 | . 2 ⊢ (∀𝑤∃*𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) |
7 | 2, 6 | sylib 219 | 1 ⊢ (∀𝑥∀𝑦∃*𝑧𝜑 → Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1526 = wceq 1528 ∃wex 1771 ∃*wmo 2613 〈cop 4563 {copab 5119 Fun wfun 6342 {coprab 7146 |
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 df-oprab 7149 |
This theorem is referenced by: funoprab 7263 fnoprabg 7264 oprabexd 7665 |
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