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Mirrors > Home > MPE Home > Th. List > r1elssi | Structured version Visualization version GIF version |
Description: The range of the 𝑅1 function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 9844 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 5280 | . . . 4 ⊢ (∀𝑥 ∈ On Tr (𝑅1‘𝑥) → Tr ∪ 𝑥 ∈ On (𝑅1‘𝑥)) | |
2 | r1tr 9814 | . . . . 5 ⊢ Tr (𝑅1‘𝑥) | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝑥 ∈ On → Tr (𝑅1‘𝑥)) |
4 | 1, 3 | mprg 3065 | . . 3 ⊢ Tr ∪ 𝑥 ∈ On (𝑅1‘𝑥) |
5 | r1funlim 9804 | . . . . . 6 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
6 | 5 | simpli 483 | . . . . 5 ⊢ Fun 𝑅1 |
7 | funiunfv 7268 | . . . . 5 ⊢ (Fun 𝑅1 → ∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ (𝑅1 “ On)) | |
8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ ∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ (𝑅1 “ On) |
9 | treq 5273 | . . . 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 5276 | . 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 1537 ∈ wcel 2106 ⊆ wss 3963 ∪ cuni 4912 ∪ ciun 4996 Tr wtr 5265 dom cdm 5689 “ cima 5692 Oncon0 6386 Lim wlim 6387 Fun wfun 6557 ‘cfv 6563 𝑅1cr1 9800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-r1 9802 |
This theorem is referenced by: r1elss 9844 pwwf 9845 rankelb 9862 rankval3b 9864 r1pw 9883 rankuni2b 9891 tcwf 9921 tcrank 9922 hsmexlem4 10467 rankcf 10815 wfgru 10854 grur1 10858 trwf 44937 wfaxsep 44951 |
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