Theorem oacomf1olem 8558

Description: Lemma for oacomf1o 8559. (Contributed by Mario Carneiro, 30-May-2015.)

Hypothesis

Ref	Expression
oacomf1olem.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))

Assertion

Ref	Expression
oacomf1olem	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:𝐴–1-1-onto→ran 𝐹 ∧ (ran 𝐹 ∩ 𝐵) = ∅))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem oacomf1olem

Step	Hyp	Ref	Expression
1		oaf1o 8557	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝑥 ∈ On ↦ (𝐵 +o 𝑥)):On–1-1-onto→(On ∖ 𝐵))
2	1	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ On ↦ (𝐵 +o 𝑥)):On–1-1-onto→(On ∖ 𝐵))
3		f1of1 6819	. . . . . 6 ⊢ ((𝑥 ∈ On ↦ (𝐵 +o 𝑥)):On–1-1-onto→(On ∖ 𝐵) → (𝑥 ∈ On ↦ (𝐵 +o 𝑥)):On–1-1→(On ∖ 𝐵))
4	2, 3	syl 18	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ On ↦ (𝐵 +o 𝑥)):On–1-1→(On ∖ 𝐵))
5		onss 7790	. . . . . 6 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
6	5	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ On)
7		f1ssres 6783	. . . . 5 ⊢ (((𝑥 ∈ On ↦ (𝐵 +o 𝑥)):On–1-1→(On ∖ 𝐵) ∧ 𝐴 ⊆ On) → ((𝑥 ∈ On ↦ (𝐵 +o 𝑥)) ↾ 𝐴):𝐴–1-1→(On ∖ 𝐵))
8	4, 6, 7	syl2anc 596	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ On ↦ (𝐵 +o 𝑥)) ↾ 𝐴):𝐴–1-1→(On ∖ 𝐵))
9	6	resmptd 6038	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ On ↦ (𝐵 +o 𝑥)) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)))
10		oacomf1olem.1	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))
11	9, 10	eqtr4di 2813	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ On ↦ (𝐵 +o 𝑥)) ↾ 𝐴) = 𝐹)
12		f1eq1 6769	. . . . 5 ⊢ (((𝑥 ∈ On ↦ (𝐵 +o 𝑥)) ↾ 𝐴) = 𝐹 → (((𝑥 ∈ On ↦ (𝐵 +o 𝑥)) ↾ 𝐴):𝐴–1-1→(On ∖ 𝐵) ↔ 𝐹:𝐴–1-1→(On ∖ 𝐵)))
13	11, 12	syl 18	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (((𝑥 ∈ On ↦ (𝐵 +o 𝑥)) ↾ 𝐴):𝐴–1-1→(On ∖ 𝐵) ↔ 𝐹:𝐴–1-1→(On ∖ 𝐵)))
14	8, 13	mpbid 235	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:𝐴–1-1→(On ∖ 𝐵))
15		f1f1orn 6832	. . 3 ⊢ (𝐹:𝐴–1-1→(On ∖ 𝐵) → 𝐹:𝐴–1-1-onto→ran 𝐹)
16	14, 15	syl 18	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:𝐴–1-1-onto→ran 𝐹)
17		f1f 6774	. . . 4 ⊢ (𝐹:𝐴–1-1→(On ∖ 𝐵) → 𝐹:𝐴⟶(On ∖ 𝐵))
18		frn 6713	. . . 4 ⊢ (𝐹:𝐴⟶(On ∖ 𝐵) → ran 𝐹 ⊆ (On ∖ 𝐵))
19	14, 17, 18	3syl 19	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ (On ∖ 𝐵))
20	19	difss2d 4086	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ On)
21		reldisj 4406	. . . 4 ⊢ (ran 𝐹 ⊆ On → ((ran 𝐹 ∩ 𝐵) = ∅ ↔ ran 𝐹 ⊆ (On ∖ 𝐵)))
22	20, 21	syl 18	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((ran 𝐹 ∩ 𝐵) = ∅ ↔ ran 𝐹 ⊆ (On ∖ 𝐵)))
23	19, 22	mpbird 260	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran 𝐹 ∩ 𝐵) = ∅)
24	16, 23	jca 521	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:𝐴–1-1-onto→ran 𝐹 ∧ (ran 𝐹 ∩ 𝐵) = ∅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279   ↦ cmpt 5186  ran crn 5656   ↾ cres 5657  Oncon0 6359  ⟶wf 6531  –1-1→wf1 6532  –1-1-onto→wf1o 6534  (class class class)co 7416   +_o coa 8459

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7742

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6301  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-oadd 8466

This theorem is used by:  oacomf1o  8559  onadju  10221