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Mirrors > Home > MPE Home > Th. List > jech9.3 | Structured version Visualization version GIF version |
Description: Every set belongs to some value of the cumulative hierarchy of sets function 𝑅1, i.e. the indexed union of all values of 𝑅1 is the universe. Lemma 9.3 of [Jech] p. 71. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 8-Jun-2013.) |
Ref | Expression |
---|---|
jech9.3 | ⊢ ∪ 𝑥 ∈ On (𝑅1‘𝑥) = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r1fnon 8907 | . . 3 ⊢ 𝑅1 Fn On | |
2 | fniunfv 6760 | . . 3 ⊢ (𝑅1 Fn On → ∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ ran 𝑅1) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ ran 𝑅1 |
4 | fndm 6223 | . . . . . 6 ⊢ (𝑅1 Fn On → dom 𝑅1 = On) | |
5 | 1, 4 | ax-mp 5 | . . . . 5 ⊢ dom 𝑅1 = On |
6 | 5 | imaeq2i 5705 | . . . 4 ⊢ (𝑅1 “ dom 𝑅1) = (𝑅1 “ On) |
7 | imadmrn 5717 | . . . 4 ⊢ (𝑅1 “ dom 𝑅1) = ran 𝑅1 | |
8 | 6, 7 | eqtr3i 2851 | . . 3 ⊢ (𝑅1 “ On) = ran 𝑅1 |
9 | 8 | unieqi 4667 | . 2 ⊢ ∪ (𝑅1 “ On) = ∪ ran 𝑅1 |
10 | unir1 8953 | . 2 ⊢ ∪ (𝑅1 “ On) = V | |
11 | 3, 9, 10 | 3eqtr2i 2855 | 1 ⊢ ∪ 𝑥 ∈ On (𝑅1‘𝑥) = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 Vcvv 3414 ∪ cuni 4658 ∪ ciun 4740 dom cdm 5342 ran crn 5343 “ cima 5345 Oncon0 5963 Fn wfn 6118 ‘cfv 6123 𝑅1cr1 8902 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-reg 8766 ax-inf2 8815 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-om 7327 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-r1 8904 |
This theorem is referenced by: (None) |
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