Theorem r1elssi 9236

Description: The range of the 𝑅₁ function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 9237 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 5187	. . . 4 ⊢ (∀𝑥 ∈ On Tr (𝑅1‘𝑥) → Tr ∪ 𝑥 ∈ On (𝑅1‘𝑥))
2		r1tr 9207	. . . . 5 ⊢ Tr (𝑅1‘𝑥)
3	2	a1i 11	. . . 4 ⊢ (𝑥 ∈ On → Tr (𝑅1‘𝑥))
4	1, 3	mprg 3154	. . 3 ⊢ Tr ∪ 𝑥 ∈ On (𝑅1‘𝑥)
5		r1funlim 9197	. . . . . 6 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
6	5	simpli 486	. . . . 5 ⊢ Fun 𝑅1
7		funiunfv 7009	. . . . 5 ⊢ (Fun 𝑅1 → ∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ (𝑅1 “ On))
8	6, 7	ax-mp 5	. . . 4 ⊢ ∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ (𝑅1 “ On)
9		treq 5180	. . . 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 232	. 2 ⊢ Tr ∪ (𝑅1 “ On)
12		trss 5183	. 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 208   = wceq 1537   ∈ wcel 2114   ⊆ wss 3938  ∪ cuni 4840  ∪ ciun 4921  Tr wtr 5174  dom cdm 5557   “ cima 5560  Oncon0 6193  Lim wlim 6194  Fun wfun 6351  ‘cfv 6357  𝑅₁cr1 9193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-r1 9195

This theorem is referenced by:  r1elss  9237  pwwf  9238  rankelb  9255  rankval3b  9257  r1pw  9276  rankuni2b  9284  tcwf  9314  tcrank  9315  hsmexlem4  9853  rankcf  10201  wfgru  10240  grur1  10244