Theorem r1elss 9730

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 9729	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On))
2		r1elss.1	. . . 4 ⊢ 𝐴 ∈ V
3	2	tz9.12 9714	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥))
4		dfss3 3924	. . . 4 ⊢ (𝐴 ⊆ ∪ (𝑅1 “ On) ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ ∪ (𝑅1 “ On))
5		r1fnon 9691	. . . . . . . 8 ⊢ 𝑅1 Fn On
6		fnfun 6600	. . . . . . . 8 ⊢ (𝑅1 Fn On → Fun 𝑅1)
7		funiunfv 7204	. . . . . . . 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 2829	. . . . . 6 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ On (𝑅1‘𝑥) ↔ 𝑦 ∈ ∪ (𝑅1 “ On))
10		eliun 4952	. . . . . 6 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ On (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥))
11	9, 10	bitr3i 277	. . . . 5 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥))
12	11	ralbii 3084	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 𝑦 ∈ ∪ (𝑅1 “ On) ↔ ∀𝑦 ∈ 𝐴 ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥))
13	4, 12	bitri 275	. . 3 ⊢ (𝐴 ⊆ ∪ (𝑅1 “ On) ↔ ∀𝑦 ∈ 𝐴 ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥))
14	8	eleq2i 2829	. . . 4 ⊢ (𝐴 ∈ ∪ 𝑥 ∈ On (𝑅1‘𝑥) ↔ 𝐴 ∈ ∪ (𝑅1 “ On))
15		eliun 4952	. . . 4 ⊢ (𝐴 ∈ ∪ 𝑥 ∈ On (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥))
16	14, 15	bitr3i 277	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥))
17	3, 13, 16	3imtr4i 292	. 2 ⊢ (𝐴 ⊆ ∪ (𝑅1 “ On) → 𝐴 ∈ ∪ (𝑅1 “ On))
18	1, 17	impbii 209	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝐴 ⊆ ∪ (𝑅1 “ On))

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3442   ⊆ wss 3903  ∪ cuni 4865  ∪ ciun 4948   “ cima 5635  Oncon0 6325  Fun wfun 6494   Fn wfn 6495  ‘cfv 6500  𝑅₁cr1 9686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-r1 9688

This theorem is referenced by:  unir1  9737  tcwf  9807  tcrank  9808  rankcf  10700  wfgru  10739  unir1regs  35313  trfr  45318  tcfr  45319  wfaxrep  45350