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Mirrors > Home > MPE Home > Th. List > r1elss | Structured version Visualization version GIF version |
Description: The range of the 𝑅1 function is transitive. Lemma 2.10 of [Kunen] p. 97. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
r1elss.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
r1elss | ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝐴 ⊆ ∪ (𝑅1 “ On)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r1elssi 9796 | . 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On)) | |
2 | r1elss.1 | . . . 4 ⊢ 𝐴 ∈ V | |
3 | 2 | tz9.12 9781 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)) |
4 | dfss3 3969 | . . . 4 ⊢ (𝐴 ⊆ ∪ (𝑅1 “ On) ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ ∪ (𝑅1 “ On)) | |
5 | r1fnon 9758 | . . . . . . . 8 ⊢ 𝑅1 Fn On | |
6 | fnfun 6646 | . . . . . . . 8 ⊢ (𝑅1 Fn On → Fun 𝑅1) | |
7 | funiunfv 7242 | . . . . . . . 8 ⊢ (Fun 𝑅1 → ∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ (𝑅1 “ On)) | |
8 | 5, 6, 7 | mp2b 10 | . . . . . . 7 ⊢ ∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ (𝑅1 “ On) |
9 | 8 | eleq2i 2826 | . . . . . 6 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ On (𝑅1‘𝑥) ↔ 𝑦 ∈ ∪ (𝑅1 “ On)) |
10 | eliun 5000 | . . . . . 6 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ On (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)) | |
11 | 9, 10 | bitr3i 277 | . . . . 5 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)) |
12 | 11 | ralbii 3094 | . . . 4 ⊢ (∀𝑦 ∈ 𝐴 𝑦 ∈ ∪ (𝑅1 “ On) ↔ ∀𝑦 ∈ 𝐴 ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)) |
13 | 4, 12 | bitri 275 | . . 3 ⊢ (𝐴 ⊆ ∪ (𝑅1 “ On) ↔ ∀𝑦 ∈ 𝐴 ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)) |
14 | 8 | eleq2i 2826 | . . . 4 ⊢ (𝐴 ∈ ∪ 𝑥 ∈ On (𝑅1‘𝑥) ↔ 𝐴 ∈ ∪ (𝑅1 “ On)) |
15 | eliun 5000 | . . . 4 ⊢ (𝐴 ∈ ∪ 𝑥 ∈ On (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)) | |
16 | 14, 15 | bitr3i 277 | . . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)) |
17 | 3, 13, 16 | 3imtr4i 292 | . 2 ⊢ (𝐴 ⊆ ∪ (𝑅1 “ On) → 𝐴 ∈ ∪ (𝑅1 “ On)) |
18 | 1, 17 | impbii 208 | 1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝐴 ⊆ ∪ (𝑅1 “ On)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ⊆ wss 3947 ∪ cuni 4907 ∪ ciun 4996 “ cima 5678 Oncon0 6361 Fun wfun 6534 Fn wfn 6535 ‘cfv 6540 𝑅1cr1 9753 |
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 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-om 7851 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-r1 9755 |
This theorem is referenced by: unir1 9804 tcwf 9874 tcrank 9875 rankcf 10768 wfgru 10807 |
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