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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 6390 | . . 3 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} ↔ ∀𝑥∃*𝑦{𝑦 ∣ 𝜑} = {𝑦}) | |
2 | mo2icl 3705 | . . . . 5 ⊢ (∀𝑧({𝑦 ∣ 𝜑} = {𝑧} → 𝑧 = ∪ {𝑦 ∣ 𝜑}) → ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧}) | |
3 | unieq 4849 | . . . . . 6 ⊢ ({𝑦 ∣ 𝜑} = {𝑧} → ∪ {𝑦 ∣ 𝜑} = ∪ {𝑧}) | |
4 | vex 3497 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
5 | 4 | unisn 4858 | . . . . . 6 ⊢ ∪ {𝑧} = 𝑧 |
6 | 3, 5 | syl6req 2873 | . . . . 5 ⊢ ({𝑦 ∣ 𝜑} = {𝑧} → 𝑧 = ∪ {𝑦 ∣ 𝜑}) |
7 | 2, 6 | mpg 1798 | . . . 4 ⊢ ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧} |
8 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑧{𝑦 ∣ 𝜑} = {𝑦} | |
9 | nfab1 2979 | . . . . . 6 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑} | |
10 | 9 | nfeq1 2993 | . . . . 5 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑} = {𝑧} |
11 | sneq 4577 | . . . . . 6 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧}) | |
12 | 11 | eqeq2d 2832 | . . . . 5 ⊢ (𝑦 = 𝑧 → ({𝑦 ∣ 𝜑} = {𝑦} ↔ {𝑦 ∣ 𝜑} = {𝑧})) |
13 | 8, 10, 12 | cbvmow 2688 | . . . 4 ⊢ (∃*𝑦{𝑦 ∣ 𝜑} = {𝑦} ↔ ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧}) |
14 | 7, 13 | mpbir 233 | . . 3 ⊢ ∃*𝑦{𝑦 ∣ 𝜑} = {𝑦} |
15 | 1, 14 | mpgbir 1800 | . 2 ⊢ Fun {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} |
16 | opabiota.1 | . . 3 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} | |
17 | 16 | funeqi 6376 | . 2 ⊢ (Fun 𝐹 ↔ Fun {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}}) |
18 | 15, 17 | mpbir 233 | 1 ⊢ Fun 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∃*wmo 2620 {cab 2799 {csn 4567 ∪ cuni 4838 {copab 5128 Fun wfun 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-fun 6357 |
This theorem is referenced by: opabiota 6746 |
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