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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 6084 | . . 3 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} ↔ ∀𝑥∃*𝑦{𝑦 ∣ 𝜑} = {𝑦}) | |
2 | mo2icl 3526 | . . . . 5 ⊢ (∀𝑧({𝑦 ∣ 𝜑} = {𝑧} → 𝑧 = ∪ {𝑦 ∣ 𝜑}) → ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧}) | |
3 | unieq 4596 | . . . . . 6 ⊢ ({𝑦 ∣ 𝜑} = {𝑧} → ∪ {𝑦 ∣ 𝜑} = ∪ {𝑧}) | |
4 | vex 3343 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
5 | 4 | unisn 4603 | . . . . . 6 ⊢ ∪ {𝑧} = 𝑧 |
6 | 3, 5 | syl6req 2811 | . . . . 5 ⊢ ({𝑦 ∣ 𝜑} = {𝑧} → 𝑧 = ∪ {𝑦 ∣ 𝜑}) |
7 | 2, 6 | mpg 1873 | . . . 4 ⊢ ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧} |
8 | nfv 1992 | . . . . 5 ⊢ Ⅎ𝑧{𝑦 ∣ 𝜑} = {𝑦} | |
9 | nfab1 2904 | . . . . . 6 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑} | |
10 | 9 | nfeq1 2916 | . . . . 5 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑} = {𝑧} |
11 | sneq 4331 | . . . . . 6 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧}) | |
12 | 11 | eqeq2d 2770 | . . . . 5 ⊢ (𝑦 = 𝑧 → ({𝑦 ∣ 𝜑} = {𝑦} ↔ {𝑦 ∣ 𝜑} = {𝑧})) |
13 | 8, 10, 12 | cbvmo 2644 | . . . 4 ⊢ (∃*𝑦{𝑦 ∣ 𝜑} = {𝑦} ↔ ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧}) |
14 | 7, 13 | mpbir 221 | . . 3 ⊢ ∃*𝑦{𝑦 ∣ 𝜑} = {𝑦} |
15 | 1, 14 | mpgbir 1875 | . 2 ⊢ Fun {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} |
16 | opabiota.1 | . . 3 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} | |
17 | 16 | funeqi 6070 | . 2 ⊢ (Fun 𝐹 ↔ Fun {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}}) |
18 | 15, 17 | mpbir 221 | 1 ⊢ Fun 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 ∃*wmo 2608 {cab 2746 {csn 4321 ∪ cuni 4588 {copab 4864 Fun wfun 6043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-fun 6051 |
This theorem is referenced by: opabiota 6423 |
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