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Mirrors > Home > MPE Home > Th. List > opabiotafun | Structured version Visualization version GIF version |
Description: Define a function whose value is "the unique 𝑦 such that 𝜑(𝑥, 𝑦)". (Contributed by NM, 19-May-2015.) |
Ref | Expression |
---|---|
opabiota.1 | ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} |
Ref | Expression |
---|---|
opabiotafun | ⊢ Fun 𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funopab 6359 | . . 3 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} ↔ ∀𝑥∃*𝑦{𝑦 ∣ 𝜑} = {𝑦}) | |
2 | mo2icl 3653 | . . . . 5 ⊢ (∀𝑧({𝑦 ∣ 𝜑} = {𝑧} → 𝑧 = ∪ {𝑦 ∣ 𝜑}) → ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧}) | |
3 | unieq 4811 | . . . . . 6 ⊢ ({𝑦 ∣ 𝜑} = {𝑧} → ∪ {𝑦 ∣ 𝜑} = ∪ {𝑧}) | |
4 | vex 3444 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
5 | 4 | unisn 4820 | . . . . . 6 ⊢ ∪ {𝑧} = 𝑧 |
6 | 3, 5 | eqtr2di 2850 | . . . . 5 ⊢ ({𝑦 ∣ 𝜑} = {𝑧} → 𝑧 = ∪ {𝑦 ∣ 𝜑}) |
7 | 2, 6 | mpg 1799 | . . . 4 ⊢ ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧} |
8 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑧{𝑦 ∣ 𝜑} = {𝑦} | |
9 | nfab1 2957 | . . . . . 6 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑} | |
10 | 9 | nfeq1 2970 | . . . . 5 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑} = {𝑧} |
11 | sneq 4535 | . . . . . 6 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧}) | |
12 | 11 | eqeq2d 2809 | . . . . 5 ⊢ (𝑦 = 𝑧 → ({𝑦 ∣ 𝜑} = {𝑦} ↔ {𝑦 ∣ 𝜑} = {𝑧})) |
13 | 8, 10, 12 | cbvmow 2663 | . . . 4 ⊢ (∃*𝑦{𝑦 ∣ 𝜑} = {𝑦} ↔ ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧}) |
14 | 7, 13 | mpbir 234 | . . 3 ⊢ ∃*𝑦{𝑦 ∣ 𝜑} = {𝑦} |
15 | 1, 14 | mpgbir 1801 | . 2 ⊢ Fun {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} |
16 | opabiota.1 | . . 3 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} | |
17 | 16 | funeqi 6345 | . 2 ⊢ (Fun 𝐹 ↔ Fun {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}}) |
18 | 15, 17 | mpbir 234 | 1 ⊢ Fun 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∃*wmo 2596 {cab 2776 {csn 4525 ∪ cuni 4800 {copab 5092 Fun wfun 6318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-fun 6326 |
This theorem is referenced by: opabiota 6721 |
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