Theorem r1elssi 9850

Description: The range of the 𝑅₁ function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 9851 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 5233	. . . 4 ⊢ (∀𝑥 ∈ On Tr (𝑅1‘𝑥) → Tr ∪ 𝑥 ∈ On (𝑅1‘𝑥))
2		r1tr 9820	. . . . 5 ⊢ Tr (𝑅1‘𝑥)
3	2	a1i 11	. . . 4 ⊢ (𝑥 ∈ On → Tr (𝑅1‘𝑥))
4	1, 3	mprg 3086	. . 3 ⊢ Tr ∪ 𝑥 ∈ On (𝑅1‘𝑥)
5		r1fun 9808	. . . . 5 ⊢ Fun 𝑅1
6		funiunfv 7283	. . . . 5 ⊢ (Fun 𝑅1 → ∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ (𝑅1 “ On))
7	5, 6	ax-mp 5	. . . 4 ⊢ ∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ (𝑅1 “ On)
8		treq 5225	. . . 4 ⊢ (∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ (𝑅1 “ On) → (Tr ∪ 𝑥 ∈ On (𝑅1‘𝑥) ↔ Tr ∪ (𝑅1 “ On)))
9	7, 8	ax-mp 5	. . 3 ⊢ (Tr ∪ 𝑥 ∈ On (𝑅1‘𝑥) ↔ Tr ∪ (𝑅1 “ On))
10	4, 9	mpbi 233	. 2 ⊢ Tr ∪ (𝑅1 “ On)
11		trss 5228	. 2 ⊢ (Tr ∪ (𝑅1 “ On) → (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On)))
12	10, 11	ax-mp 5	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900  ∪ cuni 4872  ∪ ciun 4956  Tr wtr 5218   “ cima 5661  Oncon0 6388  Fun wfun 6560  ‘cfv 6566  𝑅₁cr1 9803

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-r1 9805

This theorem is used by:  r1elss  9851  pwwf  9852  rankelb  9870  rankval3b  9873  r1pw  9896  rankuni2b  9904  tcwf  9937  tcrank  9938  hsmexlem4  10546  rankcf  10901  wfgru  10940  grur1  10944  trwf  46213  tcfr  46217  wfaxsep  46249  wfaxpow  46251