Theorem jech9.3OLD 9816

Description: Obsolete version of jech9.3 9815 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 9766	. . 3 ⊢ 𝑅1 Fn On
2		fniunfv 7249	. . 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 6640	. . . . . 6 ⊢ (𝑅1 Fn On → dom 𝑅1 = On)
5	1, 4	ax-mp 5	. . . . 5 ⊢ dom 𝑅1 = On
6	5	imaeq2i 6050	. . . 4 ⊢ (𝑅1 “ dom 𝑅1) = (𝑅1 “ On)
7		imadmrn 6067	. . . 4 ⊢ (𝑅1 “ dom 𝑅1) = ran 𝑅1
8	6, 7	eqtr3i 2786	. . 3 ⊢ (𝑅1 “ On) = ran 𝑅1
9	8	unieqi 4879	. 2 ⊢ ∪ (𝑅1 “ On) = ∪ ran 𝑅1
10		unir1 9814	. 2 ⊢ ∪ (𝑅1 “ On) = V
11	3, 9, 10	3eqtr2i 2790	1 ⊢ ∪ 𝑥 ∈ On (𝑅1‘𝑥) = V

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3451  ∪ cuni 4867  ∪ ciun 4951  dom cdm 5651  ran crn 5652   “ cima 5654  Oncon0 6361   Fn wfn 6532  ‘cfv 6537  𝑅₁cr1 9759

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-reg 9579  ax-inf2 9635

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  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 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-om 7876  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-r1 9761

This theorem is used by: (None)