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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 9184 | . . 3 ⊢ 𝑅1 Fn On | |
2 | fniunfv 6997 | . . 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 6448 | . . . . . 6 ⊢ (𝑅1 Fn On → dom 𝑅1 = On) | |
5 | 1, 4 | ax-mp 5 | . . . . 5 ⊢ dom 𝑅1 = On |
6 | 5 | imaeq2i 5920 | . . . 4 ⊢ (𝑅1 “ dom 𝑅1) = (𝑅1 “ On) |
7 | imadmrn 5932 | . . . 4 ⊢ (𝑅1 “ dom 𝑅1) = ran 𝑅1 | |
8 | 6, 7 | eqtr3i 2843 | . . 3 ⊢ (𝑅1 “ On) = ran 𝑅1 |
9 | 8 | unieqi 4839 | . 2 ⊢ ∪ (𝑅1 “ On) = ∪ ran 𝑅1 |
10 | unir1 9230 | . 2 ⊢ ∪ (𝑅1 “ On) = V | |
11 | 3, 9, 10 | 3eqtr2i 2847 | 1 ⊢ ∪ 𝑥 ∈ On (𝑅1‘𝑥) = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 Vcvv 3492 ∪ cuni 4830 ∪ ciun 4910 dom cdm 5548 ran crn 5549 “ cima 5551 Oncon0 6184 Fn wfn 6343 ‘cfv 6348 𝑅1cr1 9179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-reg 9044 ax-inf2 9092 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-r1 9181 |
This theorem is referenced by: (None) |
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