Theorem r1elss 9758

Description: The range of the 𝑅₁ function is transitive. Lemma 2.10 of [Kunen] p. 97. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)

Hypothesis

Ref	Expression
r1elss.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
r1elss	⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝐴 ⊆ ∪ (𝑅1 “ On))

Proof of Theorem r1elss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		r1elssi 9757	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On))
2		r1elss.1	. . . 4 ⊢ 𝐴 ∈ V
3	2	tz9.12 9742	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥))
4		dfss3 3923	. . . 4 ⊢ (𝐴 ⊆ ∪ (𝑅1 “ On) ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ ∪ (𝑅1 “ On))
5		r1fnon 9719	. . . . . . . 8 ⊢ 𝑅1 Fn On
6		fnfun 6616	. . . . . . . 8 ⊢ (𝑅1 Fn On → Fun 𝑅1)
7		funiunfv 7227	. . . . . . . 8 ⊢ (Fun 𝑅1 → ∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ (𝑅1 “ On))
8	5, 6, 7	mp2b 10	. . . . . . 7 ⊢ ∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ (𝑅1 “ On)
9	8	eleq2i 2853	. . . . . 6 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ On (𝑅1‘𝑥) ↔ 𝑦 ∈ ∪ (𝑅1 “ On))
10		eliun 4950	. . . . . 6 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ On (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥))
11	9, 10	bitr3i 279	. . . . 5 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥))
12	11	ralbii 3107	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 𝑦 ∈ ∪ (𝑅1 “ On) ↔ ∀𝑦 ∈ 𝐴 ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥))
13	4, 12	bitri 277	. . 3 ⊢ (𝐴 ⊆ ∪ (𝑅1 “ On) ↔ ∀𝑦 ∈ 𝐴 ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥))
14	8	eleq2i 2853	. . . 4 ⊢ (𝐴 ∈ ∪ 𝑥 ∈ On (𝑅1‘𝑥) ↔ 𝐴 ∈ ∪ (𝑅1 “ On))
15		eliun 4950	. . . 4 ⊢ (𝐴 ∈ ∪ 𝑥 ∈ On (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥))
16	14, 15	bitr3i 279	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥))
17	3, 13, 16	3imtr4i 294	. 2 ⊢ (𝐴 ⊆ ∪ (𝑅1 “ On) → 𝐴 ∈ ∪ (𝑅1 “ On))
18	1, 17	impbii 211	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝐴 ⊆ ∪ (𝑅1 “ On))

Syntax hints:   ↔ wb 208   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085  Vcvv 3453   ⊆ wss 3902  ∪ cuni 4862  ∪ ciun 4946   “ cima 5646  Oncon0 6341  Fun wfun 6510   Fn wfn 6511  ‘cfv 6516  𝑅₁cr1 9714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7394  df-om 7842  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-r1 9716

This theorem is referenced by:  unir1  9765  tcwf  9835  tcrank  9836  rankcf  10729  wfgru  10768  unir1regs  35392  ttcwf2  36846  trfr  45499  tcfr  45500  wfaxrep  45531