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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 6383 | . . 3 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} ↔ ∀𝑥∃*𝑦{𝑦 ∣ 𝜑} = {𝑦}) | |
2 | mo2icl 3702 | . . . . 5 ⊢ (∀𝑧({𝑦 ∣ 𝜑} = {𝑧} → 𝑧 = ∪ {𝑦 ∣ 𝜑}) → ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧}) | |
3 | unieq 4838 | . . . . . 6 ⊢ ({𝑦 ∣ 𝜑} = {𝑧} → ∪ {𝑦 ∣ 𝜑} = ∪ {𝑧}) | |
4 | vex 3495 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
5 | 4 | unisn 4846 | . . . . . 6 ⊢ ∪ {𝑧} = 𝑧 |
6 | 3, 5 | syl6req 2870 | . . . . 5 ⊢ ({𝑦 ∣ 𝜑} = {𝑧} → 𝑧 = ∪ {𝑦 ∣ 𝜑}) |
7 | 2, 6 | mpg 1789 | . . . 4 ⊢ ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧} |
8 | nfv 1906 | . . . . 5 ⊢ Ⅎ𝑧{𝑦 ∣ 𝜑} = {𝑦} | |
9 | nfab1 2976 | . . . . . 6 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑} | |
10 | 9 | nfeq1 2990 | . . . . 5 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑} = {𝑧} |
11 | sneq 4567 | . . . . . 6 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧}) | |
12 | 11 | eqeq2d 2829 | . . . . 5 ⊢ (𝑦 = 𝑧 → ({𝑦 ∣ 𝜑} = {𝑦} ↔ {𝑦 ∣ 𝜑} = {𝑧})) |
13 | 8, 10, 12 | cbvmow 2681 | . . . 4 ⊢ (∃*𝑦{𝑦 ∣ 𝜑} = {𝑦} ↔ ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧}) |
14 | 7, 13 | mpbir 232 | . . 3 ⊢ ∃*𝑦{𝑦 ∣ 𝜑} = {𝑦} |
15 | 1, 14 | mpgbir 1791 | . 2 ⊢ Fun {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} |
16 | opabiota.1 | . . 3 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} | |
17 | 16 | funeqi 6369 | . 2 ⊢ (Fun 𝐹 ↔ Fun {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}}) |
18 | 15, 17 | mpbir 232 | 1 ⊢ Fun 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∃*wmo 2613 {cab 2796 {csn 4557 ∪ cuni 4830 {copab 5119 Fun wfun 6342 |
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-rex 3141 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-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-fun 6350 |
This theorem is referenced by: opabiota 6739 |
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