Theorem jech9.3OLD 9830

Description: Obsolete version of jech9.3 9829 as of 29-Sep-2026. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 8-Jun-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem jech9.3OLD

Step	Hyp	Ref	Expression
1		r1fnon 9780	. . 3 ⊢ 𝑅1 Fn On
2		fniunfv 7255	. . 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 6647	. . . . . 6 ⊢ (𝑅1 Fn On → dom 𝑅1 = On)
5	1, 4	ax-mp 5	. . . . 5 ⊢ dom 𝑅1 = On
6	5	imaeq2i 6052	. . . 4 ⊢ (𝑅1 “ dom 𝑅1) = (𝑅1 “ On)
7		imadmrn 6069	. . . 4 ⊢ (𝑅1 “ dom 𝑅1) = ran 𝑅1
8	6, 7	eqtr3i 2785	. . 3 ⊢ (𝑅1 “ On) = ran 𝑅1
9	8	unieqi 4878	. 2 ⊢ ∪ (𝑅1 “ On) = ∪ ran 𝑅1
10		unir1 9828	. 2 ⊢ ∪ (𝑅1 “ On) = V
11	3, 9, 10	3eqtr2i 2789	1 ⊢ ∪ 𝑥 ∈ On (𝑅1‘𝑥) = V

This proof depends on syntax axioms:   = wceq 1569  Vcvv 3450  ∪ cuni 4866  ∪ ciun 4950  dom cdm 5651  ran crn 5652   “ cima 5654  Oncon0 6365   Fn wfn 6538  ‘cfv 6543  𝑅₁cr1 9773

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-reg 9593  ax-inf2 9649

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-om 7882  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-r1 9775

This theorem is used by: (None)