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Mirrors > Home > MPE Home > Th. List > axcc4dom | Structured version Visualization version GIF version |
Description: Relax the constraint on axcc4 10376 to dominance instead of equinumerosity. (Contributed by Mario Carneiro, 18-Jan-2014.) |
Ref | Expression |
---|---|
axcc4dom.1 | ⊢ 𝐴 ∈ V |
axcc4dom.2 | ⊢ (𝑥 = (𝑓‘𝑛) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
axcc4dom | ⊢ ((𝑁 ≼ ω ∧ ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜑) → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brdom2 8923 | . . 3 ⊢ (𝑁 ≼ ω ↔ (𝑁 ≺ ω ∨ 𝑁 ≈ ω)) | |
2 | isfinite 9589 | . . . . 5 ⊢ (𝑁 ∈ Fin ↔ 𝑁 ≺ ω) | |
3 | axcc4dom.2 | . . . . . . 7 ⊢ (𝑥 = (𝑓‘𝑛) → (𝜑 ↔ 𝜓)) | |
4 | 3 | ac6sfi 9232 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜑) → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜓)) |
5 | 4 | ex 414 | . . . . 5 ⊢ (𝑁 ∈ Fin → (∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜓))) |
6 | 2, 5 | sylbir 234 | . . . 4 ⊢ (𝑁 ≺ ω → (∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜓))) |
7 | raleq 3310 | . . . . . 6 ⊢ (𝑁 = if(𝑁 ≈ ω, 𝑁, ω) → (∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑛 ∈ if (𝑁 ≈ ω, 𝑁, ω)∃𝑥 ∈ 𝐴 𝜑)) | |
8 | feq2 6651 | . . . . . . . 8 ⊢ (𝑁 = if(𝑁 ≈ ω, 𝑁, ω) → (𝑓:𝑁⟶𝐴 ↔ 𝑓:if(𝑁 ≈ ω, 𝑁, ω)⟶𝐴)) | |
9 | raleq 3310 | . . . . . . . 8 ⊢ (𝑁 = if(𝑁 ≈ ω, 𝑁, ω) → (∀𝑛 ∈ 𝑁 𝜓 ↔ ∀𝑛 ∈ if (𝑁 ≈ ω, 𝑁, ω)𝜓)) | |
10 | 8, 9 | anbi12d 632 | . . . . . . 7 ⊢ (𝑁 = if(𝑁 ≈ ω, 𝑁, ω) → ((𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜓) ↔ (𝑓:if(𝑁 ≈ ω, 𝑁, ω)⟶𝐴 ∧ ∀𝑛 ∈ if (𝑁 ≈ ω, 𝑁, ω)𝜓))) |
11 | 10 | exbidv 1925 | . . . . . 6 ⊢ (𝑁 = if(𝑁 ≈ ω, 𝑁, ω) → (∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜓) ↔ ∃𝑓(𝑓:if(𝑁 ≈ ω, 𝑁, ω)⟶𝐴 ∧ ∀𝑛 ∈ if (𝑁 ≈ ω, 𝑁, ω)𝜓))) |
12 | 7, 11 | imbi12d 345 | . . . . 5 ⊢ (𝑁 = if(𝑁 ≈ ω, 𝑁, ω) → ((∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜓)) ↔ (∀𝑛 ∈ if (𝑁 ≈ ω, 𝑁, ω)∃𝑥 ∈ 𝐴 𝜑 → ∃𝑓(𝑓:if(𝑁 ≈ ω, 𝑁, ω)⟶𝐴 ∧ ∀𝑛 ∈ if (𝑁 ≈ ω, 𝑁, ω)𝜓)))) |
13 | axcc4dom.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
14 | breq1 5109 | . . . . . . 7 ⊢ (𝑁 = if(𝑁 ≈ ω, 𝑁, ω) → (𝑁 ≈ ω ↔ if(𝑁 ≈ ω, 𝑁, ω) ≈ ω)) | |
15 | breq1 5109 | . . . . . . 7 ⊢ (ω = if(𝑁 ≈ ω, 𝑁, ω) → (ω ≈ ω ↔ if(𝑁 ≈ ω, 𝑁, ω) ≈ ω)) | |
16 | omex 9580 | . . . . . . . 8 ⊢ ω ∈ V | |
17 | 16 | enref 8926 | . . . . . . 7 ⊢ ω ≈ ω |
18 | 14, 15, 17 | elimhyp 4552 | . . . . . 6 ⊢ if(𝑁 ≈ ω, 𝑁, ω) ≈ ω |
19 | 13, 18, 3 | axcc4 10376 | . . . . 5 ⊢ (∀𝑛 ∈ if (𝑁 ≈ ω, 𝑁, ω)∃𝑥 ∈ 𝐴 𝜑 → ∃𝑓(𝑓:if(𝑁 ≈ ω, 𝑁, ω)⟶𝐴 ∧ ∀𝑛 ∈ if (𝑁 ≈ ω, 𝑁, ω)𝜓)) |
20 | 12, 19 | dedth 4545 | . . . 4 ⊢ (𝑁 ≈ ω → (∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜓))) |
21 | 6, 20 | jaoi 856 | . . 3 ⊢ ((𝑁 ≺ ω ∨ 𝑁 ≈ ω) → (∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜓))) |
22 | 1, 21 | sylbi 216 | . 2 ⊢ (𝑁 ≼ ω → (∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜓))) |
23 | 22 | imp 408 | 1 ⊢ ((𝑁 ≼ ω ∧ ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜑) → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∀wral 3065 ∃wrex 3074 Vcvv 3446 ifcif 4487 class class class wbr 5106 ⟶wf 6493 ‘cfv 6497 ωcom 7803 ≈ cen 8881 ≼ cdom 8882 ≺ csdm 8883 Fincfn 8884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9578 ax-cc 10372 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 |
This theorem is referenced by: 2ndcctbss 22809 2ndcsep 22813 iscmet3 24660 heiborlem3 36275 |
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