Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ttukeylem4 | Structured version Visualization version GIF version |
Description: Lemma for ttukey 9223. (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 5695 | . . 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 9216 | . . 3 ⊢ ((𝜑 ∧ ∅ ∈ On) → (𝐺‘∅) = if(∅ = ∪ ∅, if(∅ = ∅, 𝐵, ∪ (𝐺 “ ∅)), ((𝐺‘∪ ∅) ∪ if(((𝐺‘∪ ∅) ∪ {(𝐹‘∪ ∅)}) ∈ 𝐴, {(𝐹‘∪ ∅)}, ∅)))) |
7 | 1, 6 | mpan2 703 | . 2 ⊢ (𝜑 → (𝐺‘∅) = if(∅ = ∪ ∅, if(∅ = ∅, 𝐵, ∪ (𝐺 “ ∅)), ((𝐺‘∪ ∅) ∪ if(((𝐺‘∪ ∅) ∪ {(𝐹‘∪ ∅)}) ∈ 𝐴, {(𝐹‘∪ ∅)}, ∅)))) |
8 | uni0 4401 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
9 | 8 | eqcomi 2619 | . . . 4 ⊢ ∅ = ∪ ∅ |
10 | 9 | iftruei 4043 | . . 3 ⊢ if(∅ = ∪ ∅, if(∅ = ∅, 𝐵, ∪ (𝐺 “ ∅)), ((𝐺‘∪ ∅) ∪ if(((𝐺‘∪ ∅) ∪ {(𝐹‘∪ ∅)}) ∈ 𝐴, {(𝐹‘∪ ∅)}, ∅))) = if(∅ = ∅, 𝐵, ∪ (𝐺 “ ∅)) |
11 | eqid 2610 | . . . 4 ⊢ ∅ = ∅ | |
12 | 11 | iftruei 4043 | . . 3 ⊢ if(∅ = ∅, 𝐵, ∪ (𝐺 “ ∅)) = 𝐵 |
13 | 10, 12 | eqtri 2632 | . 2 ⊢ if(∅ = ∪ ∅, if(∅ = ∅, 𝐵, ∪ (𝐺 “ ∅)), ((𝐺‘∪ ∅) ∪ if(((𝐺‘∪ ∅) ∪ {(𝐹‘∪ ∅)}) ∈ 𝐴, {(𝐹‘∪ ∅)}, ∅))) = 𝐵 |
14 | 7, 13 | syl6eq 2660 | 1 ⊢ (𝜑 → (𝐺‘∅) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ifcif 4036 𝒫 cpw 4108 {csn 4125 ∪ cuni 4372 ↦ cmpt 4643 dom cdm 5038 ran crn 5039 “ cima 5041 Oncon0 5640 –1-1-onto→wf1o 5803 ‘cfv 5804 recscrecs 7354 Fincfn 7841 cardccrd 8644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-wrecs 7294 df-recs 7355 |
This theorem is referenced by: ttukeylem7 9220 |
Copyright terms: Public domain | W3C validator |