Theorem r1elssi 9562

Description: The range of the 𝑅₁ function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 9563 that doesn't need 𝐴 to be a set. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)

Assertion

Ref	Expression
r1elssi	⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On))

Proof of Theorem r1elssi

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		triun 5209	. . . 4 ⊢ (∀𝑥 ∈ On Tr (𝑅1‘𝑥) → Tr ∪ 𝑥 ∈ On (𝑅1‘𝑥))
2		r1tr 9533	. . . . 5 ⊢ Tr (𝑅1‘𝑥)
3	2	a1i 11	. . . 4 ⊢ (𝑥 ∈ On → Tr (𝑅1‘𝑥))
4	1, 3	mprg 3080	. . 3 ⊢ Tr ∪ 𝑥 ∈ On (𝑅1‘𝑥)
5		r1funlim 9523	. . . . . 6 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
6	5	simpli 484	. . . . 5 ⊢ Fun 𝑅1
7		funiunfv 7116	. . . . 5 ⊢ (Fun 𝑅1 → ∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ (𝑅1 “ On))
8	6, 7	ax-mp 5	. . . 4 ⊢ ∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ (𝑅1 “ On)
9		treq 5202	. . . 4 ⊢ (∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ (𝑅1 “ On) → (Tr ∪ 𝑥 ∈ On (𝑅1‘𝑥) ↔ Tr ∪ (𝑅1 “ On)))
10	8, 9	ax-mp 5	. . 3 ⊢ (Tr ∪ 𝑥 ∈ On (𝑅1‘𝑥) ↔ Tr ∪ (𝑅1 “ On))
11	4, 10	mpbi 229	. 2 ⊢ Tr ∪ (𝑅1 “ On)
12		trss 5205	. 2 ⊢ (Tr ∪ (𝑅1 “ On) → (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On)))
13	11, 12	ax-mp 5	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2110   ⊆ wss 3892  ∪ cuni 4845  ∪ ciun 4930  Tr wtr 5196  dom cdm 5589   “ cima 5592  Oncon0 6264  Lim wlim 6265  Fun wfun 6425  ‘cfv 6431  𝑅₁cr1 9519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6200  df-ord 6267  df-on 6268  df-lim 6269  df-suc 6270  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-ov 7272  df-om 7705  df-2nd 7823  df-frecs 8086  df-wrecs 8117  df-recs 8191  df-rdg 8230  df-r1 9521

This theorem is referenced by:  r1elss  9563  pwwf  9564  rankelb  9581  rankval3b  9583  r1pw  9602  rankuni2b  9610  tcwf  9640  tcrank  9641  hsmexlem4  10184  rankcf  10532  wfgru  10571  grur1  10575