Theorem r1elss 9721

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 9720	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On))
2		r1elss.1	. . . 4 ⊢ 𝐴 ∈ V
3	2	tz9.12 9705	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥))
4		dfss3 3911	. . . 4 ⊢ (𝐴 ⊆ ∪ (𝑅1 “ On) ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ ∪ (𝑅1 “ On))
5		r1fnon 9682	. . . . . . . 8 ⊢ 𝑅1 Fn On
6		fnfun 6592	. . . . . . . 8 ⊢ (𝑅1 Fn On → Fun 𝑅1)
7		funiunfv 7196	. . . . . . . 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 4938	. . . . . 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 4938	. . . 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 3430   ⊆ wss 3890  ∪ cuni 4851  ∪ ciun 4934   “ cima 5627  Oncon0 6317  Fun wfun 6486   Fn wfn 6487  ‘cfv 6492  𝑅₁cr1 9677

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

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 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-r1 9679

This theorem is referenced by:  unir1  9728  tcwf  9798  tcrank  9799  rankcf  10691  wfgru  10730  unir1regs  35295  ttcwf2  36723  trfr  45407  tcfr  45408  wfaxrep  45439