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Mirrors > Home > MPE Home > Th. List > ttukeylem4 | Structured version Visualization version GIF version |
Description: Lemma for ttukey 9934. (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 6239 | . . 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 9927 | . . 3 ⊢ ((𝜑 ∧ ∅ ∈ On) → (𝐺‘∅) = if(∅ = ∪ ∅, if(∅ = ∅, 𝐵, ∪ (𝐺 “ ∅)), ((𝐺‘∪ ∅) ∪ if(((𝐺‘∪ ∅) ∪ {(𝐹‘∪ ∅)}) ∈ 𝐴, {(𝐹‘∪ ∅)}, ∅)))) |
7 | 1, 6 | mpan2 689 | . 2 ⊢ (𝜑 → (𝐺‘∅) = if(∅ = ∪ ∅, if(∅ = ∅, 𝐵, ∪ (𝐺 “ ∅)), ((𝐺‘∪ ∅) ∪ if(((𝐺‘∪ ∅) ∪ {(𝐹‘∪ ∅)}) ∈ 𝐴, {(𝐹‘∪ ∅)}, ∅)))) |
8 | uni0 4859 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
9 | 8 | eqcomi 2830 | . . . 4 ⊢ ∅ = ∪ ∅ |
10 | 9 | iftruei 4474 | . . 3 ⊢ if(∅ = ∪ ∅, if(∅ = ∅, 𝐵, ∪ (𝐺 “ ∅)), ((𝐺‘∪ ∅) ∪ if(((𝐺‘∪ ∅) ∪ {(𝐹‘∪ ∅)}) ∈ 𝐴, {(𝐹‘∪ ∅)}, ∅))) = if(∅ = ∅, 𝐵, ∪ (𝐺 “ ∅)) |
11 | eqid 2821 | . . . 4 ⊢ ∅ = ∅ | |
12 | 11 | iftruei 4474 | . . 3 ⊢ if(∅ = ∅, 𝐵, ∪ (𝐺 “ ∅)) = 𝐵 |
13 | 10, 12 | eqtri 2844 | . 2 ⊢ if(∅ = ∪ ∅, if(∅ = ∅, 𝐵, ∪ (𝐺 “ ∅)), ((𝐺‘∪ ∅) ∪ if(((𝐺‘∪ ∅) ∪ {(𝐹‘∪ ∅)}) ∈ 𝐴, {(𝐹‘∪ ∅)}, ∅))) = 𝐵 |
14 | 7, 13 | syl6eq 2872 | 1 ⊢ (𝜑 → (𝐺‘∅) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1531 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ∖ cdif 3933 ∪ cun 3934 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 ifcif 4467 𝒫 cpw 4539 {csn 4561 ∪ cuni 4832 ↦ cmpt 5139 dom cdm 5550 ran crn 5551 “ cima 5553 Oncon0 6186 –1-1-onto→wf1o 6349 ‘cfv 6350 recscrecs 8001 Fincfn 8503 cardccrd 9358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-wrecs 7941 df-recs 8002 |
This theorem is referenced by: ttukeylem7 9931 |
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