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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 9842 | . 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On)) | |
2 | r1elss.1 | . . . 4 ⊢ 𝐴 ∈ V | |
3 | 2 | tz9.12 9827 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)) |
4 | dfss3 3983 | . . . 4 ⊢ (𝐴 ⊆ ∪ (𝑅1 “ On) ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ ∪ (𝑅1 “ On)) | |
5 | r1fnon 9804 | . . . . . . . 8 ⊢ 𝑅1 Fn On | |
6 | fnfun 6668 | . . . . . . . 8 ⊢ (𝑅1 Fn On → Fun 𝑅1) | |
7 | funiunfv 7267 | . . . . . . . 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 2830 | . . . . . 6 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ On (𝑅1‘𝑥) ↔ 𝑦 ∈ ∪ (𝑅1 “ On)) |
10 | eliun 4999 | . . . . . 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 3090 | . . . 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 2830 | . . . 4 ⊢ (𝐴 ∈ ∪ 𝑥 ∈ On (𝑅1‘𝑥) ↔ 𝐴 ∈ ∪ (𝑅1 “ On)) |
15 | eliun 4999 | . . . 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 209 | 1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝐴 ⊆ ∪ (𝑅1 “ On)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1536 ∈ wcel 2105 ∀wral 3058 ∃wrex 3067 Vcvv 3477 ⊆ wss 3962 ∪ cuni 4911 ∪ ciun 4995 “ cima 5691 Oncon0 6385 Fun wfun 6556 Fn wfn 6557 ‘cfv 6562 𝑅1cr1 9799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-r1 9801 |
This theorem is referenced by: unir1 9850 tcwf 9920 tcrank 9921 rankcf 10814 wfgru 10853 wfaxrep 44949 |
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