Theorem unir1 3511

Description: The cumulative hierarchy of sets covers the universe. Proposition 4.45 (b) to (a) of [Mendelson] p. 281.

Assertion

Ref	Expression
unir1	⊢ ∪(R1 “ On) = V

Proof of Theorem unir1

Step	Hyp	Ref	Expression
1		r1fnon 3494	. . . . . . 7 ⊢ R1 Fn On
2		fndm 2723	. . . . . . 7 ⊢ (R1 Fn On → dom R1 = On)
3	1, 2	ax-mp 6	. . . . . 6 ⊢ dom R1 = On
4		imaeq2 2603	. . . . . 6 ⊢ (dom R1 = On → (R1 “ dom R1) = (R1 “ On))
5	3, 4	ax-mp 6	. . . . 5 ⊢ (R1 “ dom R1) = (R1 “ On)
6		imadmrn 2610	. . . . 5 ⊢ (R1 “ dom R1) = ran R1
7	5, 6	eqtr3 1121	. . . 4 ⊢ (R1 “ On) = ran R1
8	7	unieqi 1928	. . 3 ⊢ ∪(R1 “ On) = ∪ran R1
9		fniunfv 2860	. . . 4 ⊢ (R1 Fn On → ∪x ∈ On (R1 ‘x) = ∪ran R1)
10	1, 9	ax-mp 6	. . 3 ⊢ ∪x ∈ On (R1 ‘x) = ∪ran R1
11	8, 10	eqtr4 1122	. 2 ⊢ ∪(R1 “ On) = ∪x ∈ On (R1 ‘x)
12		jech9.3 3510	. 2 ⊢ ∪x ∈ On (R1 ‘x) = V
13	11, 12	eqtr 1119	1 ⊢ ∪(R1 “ On) = V

Syntax hints:   = wceq 1091  Vcvv 1348  ∪cuni 1919  ∪ciun 1994  Oncon0 2199  dom cdm 2410  ran crn 2411   “ cima 2413   Fn wfn 2417   ‘cfv 2422  R₁cr1 3485

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-r1 3487

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