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| Description: The range of the 𝑅1 function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 9847 that doesn't need 𝐴 to be a set. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) | 
| Ref | Expression | 
|---|---|
| r1elssi | ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | triun 5273 | . . . 4 ⊢ (∀𝑥 ∈ On Tr (𝑅1‘𝑥) → Tr ∪ 𝑥 ∈ On (𝑅1‘𝑥)) | |
| 2 | r1tr 9817 | . . . . 5 ⊢ Tr (𝑅1‘𝑥) | |
| 3 | 2 | a1i 11 | . . . 4 ⊢ (𝑥 ∈ On → Tr (𝑅1‘𝑥)) | 
| 4 | 1, 3 | mprg 3066 | . . 3 ⊢ Tr ∪ 𝑥 ∈ On (𝑅1‘𝑥) | 
| 5 | r1funlim 9807 | . . . . . 6 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
| 6 | 5 | simpli 483 | . . . . 5 ⊢ Fun 𝑅1 | 
| 7 | funiunfv 7269 | . . . . 5 ⊢ (Fun 𝑅1 → ∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ (𝑅1 “ On)) | |
| 8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ ∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ (𝑅1 “ On) | 
| 9 | treq 5266 | . . . 4 ⊢ (∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ (𝑅1 “ On) → (Tr ∪ 𝑥 ∈ On (𝑅1‘𝑥) ↔ Tr ∪ (𝑅1 “ On))) | |
| 10 | 8, 9 | ax-mp 5 | . . 3 ⊢ (Tr ∪ 𝑥 ∈ On (𝑅1‘𝑥) ↔ Tr ∪ (𝑅1 “ On)) | 
| 11 | 4, 10 | mpbi 230 | . 2 ⊢ Tr ∪ (𝑅1 “ On) | 
| 12 | trss 5269 | . 2 ⊢ (Tr ∪ (𝑅1 “ On) → (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On))) | |
| 13 | 11, 12 | ax-mp 5 | 1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 ∪ cuni 4906 ∪ ciun 4990 Tr wtr 5258 dom cdm 5684 “ cima 5687 Oncon0 6383 Lim wlim 6384 Fun wfun 6554 ‘cfv 6560 𝑅1cr1 9803 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-r1 9805 | 
| This theorem is referenced by: r1elss 9847 pwwf 9848 rankelb 9865 rankval3b 9867 r1pw 9886 rankuni2b 9894 tcwf 9924 tcrank 9925 hsmexlem4 10470 rankcf 10818 wfgru 10857 grur1 10861 trwf 44981 tcfr 44985 wfaxsep 45017 wfaxpow 45019 | 
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