Theorem r1elssi 9773

Description: The range of the 𝑅₁ function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 9774 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 9744	. . . . 5 ⊢ Tr (𝑅1‘𝑥)
3	2	a1i 11	. . . 4 ⊢ (𝑥 ∈ On → Tr (𝑅1‘𝑥))
4	1, 3	mprg 3085	. . 3 ⊢ Tr ∪ 𝑥 ∈ On (𝑅1‘𝑥)
5		r1funlim 9734	. . . . . 6 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
6	5	simpli 488	. . . . 5 ⊢ Fun 𝑅1
7		funiunfv 7246	. . . . 5 ⊢ (Fun 𝑅1 → ∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ (𝑅1 “ On))
8	6, 7	ax-mp 5	. . . 4 ⊢ ∪ 𝑥 ∈ On (𝑅1‘𝑥) = ∪ (𝑅1 “ On)
9		treq 5225	. . . 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 233	. 2 ⊢ Tr ∪ (𝑅1 “ On)
12		trss 5228	. 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 209   = wceq 1570   ∈ wcel 2143   ⊆ wss 3905  ∪ cuni 4872  ∪ ciun 4956  Tr wtr 5218  dom cdm 5661   “ cima 5664  Oncon0 6360  Lim wlim 6361  Fun wfun 6530  ‘cfv 6536  𝑅₁cr1 9730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-r1 9732

This theorem is referenced by:  r1elss  9774  pwwf  9775  rankelb  9792  rankval3b  9794  r1pw  9813  rankuni2b  9821  tcwf  9851  tcrank  9852  hsmexlem4  10408  rankcf  10757  wfgru  10796  grur1  10800  trwf  45688  tcfr  45692  wfaxsep  45724  wfaxpow  45726