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| Description: Lemma for ttukey 10559. (Contributed by Mario Carneiro, 15-May-2015.) | 
| Ref | Expression | 
|---|---|
| ttukeylem.1 | ⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵)) | 
| ttukeylem.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐴) | 
| ttukeylem.3 | ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) | 
| ttukeylem.4 | ⊢ 𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = ∪ dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ∪ ran 𝑧), ((𝑧‘∪ dom 𝑧) ∪ if(((𝑧‘∪ dom 𝑧) ∪ {(𝐹‘∪ dom 𝑧)}) ∈ 𝐴, {(𝐹‘∪ dom 𝑧)}, ∅))))) | 
| Ref | Expression | 
|---|---|
| ttukeylem4 | ⊢ (𝜑 → (𝐺‘∅) = 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 0elon 6437 | . . 3 ⊢ ∅ ∈ On | |
| 2 | ttukeylem.1 | . . . 4 ⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵)) | |
| 3 | ttukeylem.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
| 4 | ttukeylem.3 | . . . 4 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) | |
| 5 | ttukeylem.4 | . . . 4 ⊢ 𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = ∪ dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ∪ ran 𝑧), ((𝑧‘∪ dom 𝑧) ∪ if(((𝑧‘∪ dom 𝑧) ∪ {(𝐹‘∪ dom 𝑧)}) ∈ 𝐴, {(𝐹‘∪ dom 𝑧)}, ∅))))) | |
| 6 | 2, 3, 4, 5 | ttukeylem3 10552 | . . 3 ⊢ ((𝜑 ∧ ∅ ∈ On) → (𝐺‘∅) = if(∅ = ∪ ∅, if(∅ = ∅, 𝐵, ∪ (𝐺 “ ∅)), ((𝐺‘∪ ∅) ∪ if(((𝐺‘∪ ∅) ∪ {(𝐹‘∪ ∅)}) ∈ 𝐴, {(𝐹‘∪ ∅)}, ∅)))) | 
| 7 | 1, 6 | mpan2 691 | . 2 ⊢ (𝜑 → (𝐺‘∅) = if(∅ = ∪ ∅, if(∅ = ∅, 𝐵, ∪ (𝐺 “ ∅)), ((𝐺‘∪ ∅) ∪ if(((𝐺‘∪ ∅) ∪ {(𝐹‘∪ ∅)}) ∈ 𝐴, {(𝐹‘∪ ∅)}, ∅)))) | 
| 8 | uni0 4934 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
| 9 | 8 | eqcomi 2745 | . . . 4 ⊢ ∅ = ∪ ∅ | 
| 10 | 9 | iftruei 4531 | . . 3 ⊢ if(∅ = ∪ ∅, if(∅ = ∅, 𝐵, ∪ (𝐺 “ ∅)), ((𝐺‘∪ ∅) ∪ if(((𝐺‘∪ ∅) ∪ {(𝐹‘∪ ∅)}) ∈ 𝐴, {(𝐹‘∪ ∅)}, ∅))) = if(∅ = ∅, 𝐵, ∪ (𝐺 “ ∅)) | 
| 11 | eqid 2736 | . . . 4 ⊢ ∅ = ∅ | |
| 12 | 11 | iftruei 4531 | . . 3 ⊢ if(∅ = ∅, 𝐵, ∪ (𝐺 “ ∅)) = 𝐵 | 
| 13 | 10, 12 | eqtri 2764 | . 2 ⊢ if(∅ = ∪ ∅, if(∅ = ∅, 𝐵, ∪ (𝐺 “ ∅)), ((𝐺‘∪ ∅) ∪ if(((𝐺‘∪ ∅) ∪ {(𝐹‘∪ ∅)}) ∈ 𝐴, {(𝐹‘∪ ∅)}, ∅))) = 𝐵 | 
| 14 | 7, 13 | eqtrdi 2792 | 1 ⊢ (𝜑 → (𝐺‘∅) = 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1537 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∖ cdif 3947 ∪ cun 3948 ∩ cin 3949 ⊆ wss 3950 ∅c0 4332 ifcif 4524 𝒫 cpw 4599 {csn 4625 ∪ cuni 4906 ↦ cmpt 5224 dom cdm 5684 ran crn 5685 “ cima 5687 Oncon0 6383 –1-1-onto→wf1o 6559 ‘cfv 6560 recscrecs 8411 Fincfn 8986 cardccrd 9976 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 | 
| This theorem is referenced by: ttukeylem7 10556 | 
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