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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 6583 | . . 3 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}} ↔ ∀𝑥∃*𝑦{𝑦 ∣ 𝜑} = {𝑦}) | |
2 | mo2icl 3710 | . . . . 5 ⊢ (∀𝑧({𝑦 ∣ 𝜑} = {𝑧} → 𝑧 = ∪ {𝑦 ∣ 𝜑}) → ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧}) | |
3 | unieq 4919 | . . . . . 6 ⊢ ({𝑦 ∣ 𝜑} = {𝑧} → ∪ {𝑦 ∣ 𝜑} = ∪ {𝑧}) | |
4 | unisnv 4931 | . . . . . 6 ⊢ ∪ {𝑧} = 𝑧 | |
5 | 3, 4 | eqtr2di 2788 | . . . . 5 ⊢ ({𝑦 ∣ 𝜑} = {𝑧} → 𝑧 = ∪ {𝑦 ∣ 𝜑}) |
6 | 2, 5 | mpg 1798 | . . . 4 ⊢ ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧} |
7 | nfv 1916 | . . . . 5 ⊢ Ⅎ𝑧{𝑦 ∣ 𝜑} = {𝑦} | |
8 | nfab1 2904 | . . . . . 6 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑} | |
9 | 8 | nfeq1 2917 | . . . . 5 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑} = {𝑧} |
10 | sneq 4638 | . . . . . 6 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧}) | |
11 | 10 | eqeq2d 2742 | . . . . 5 ⊢ (𝑦 = 𝑧 → ({𝑦 ∣ 𝜑} = {𝑦} ↔ {𝑦 ∣ 𝜑} = {𝑧})) |
12 | 7, 9, 11 | cbvmow 2596 | . . . 4 ⊢ (∃*𝑦{𝑦 ∣ 𝜑} = {𝑦} ↔ ∃*𝑧{𝑦 ∣ 𝜑} = {𝑧}) |
13 | 6, 12 | mpbir 230 | . . 3 ⊢ ∃*𝑦{𝑦 ∣ 𝜑} = {𝑦} |
14 | 1, 13 | mpgbir 1800 | . 2 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}} |
15 | opabiota.1 | . . 3 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}} | |
16 | 15 | funeqi 6569 | . 2 ⊢ (Fun 𝐹 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}) |
17 | 14, 16 | mpbir 230 | 1 ⊢ Fun 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∃*wmo 2531 {cab 2708 {csn 4628 ∪ cuni 4908 {copab 5210 Fun wfun 6537 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-fun 6545 |
This theorem is referenced by: opabiota 6974 |
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