Theorem jech9.3 9726

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.)

Assertion

Ref	Expression
jech9.3	⊢ ∪ 𝑥 ∈ On (𝑅1‘𝑥) = V

Proof of Theorem jech9.3

Step	Hyp	Ref	Expression
1		r1fnon 9679	. . 3 ⊢ 𝑅1 Fn On
2		fniunfv 7193	. . 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 6595	. . . . . 6 ⊢ (𝑅1 Fn On → dom 𝑅1 = On)
5	1, 4	ax-mp 5	. . . . 5 ⊢ dom 𝑅1 = On
6	5	imaeq2i 6017	. . . 4 ⊢ (𝑅1 “ dom 𝑅1) = (𝑅1 “ On)
7		imadmrn 6029	. . . 4 ⊢ (𝑅1 “ dom 𝑅1) = ran 𝑅1
8	6, 7	eqtr3i 2761	. . 3 ⊢ (𝑅1 “ On) = ran 𝑅1
9	8	unieqi 4875	. 2 ⊢ ∪ (𝑅1 “ On) = ∪ ran 𝑅1
10		unir1 9725	. 2 ⊢ ∪ (𝑅1 “ On) = V
11	3, 9, 10	3eqtr2i 2765	1 ⊢ ∪ 𝑥 ∈ On (𝑅1‘𝑥) = V

Syntax hints:   = wceq 1541  Vcvv 3440  ∪ cuni 4863  ∪ ciun 4946  dom cdm 5624  ran crn 5625   “ cima 5627  Oncon0 6317   Fn wfn 6487  ‘cfv 6492  𝑅₁cr1 9674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-reg 9497  ax-inf2 9550

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-r1 9676

This theorem is referenced by: (None)