Theorem oacomf1olem 8281

Description: Lemma for oacomf1o 8282. (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 8280	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝑥 ∈ On ↦ (𝐵 +o 𝑥)):On–1-1-onto→(On ∖ 𝐵))
2	1	adantl 485	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ On ↦ (𝐵 +o 𝑥)):On–1-1-onto→(On ∖ 𝐵))
3		f1of1 6649	. . . . . 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 17	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ On ↦ (𝐵 +o 𝑥)):On–1-1→(On ∖ 𝐵))
5		onss 7557	. . . . . 6 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
6	5	adantr 484	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ On)
7		f1ssres 6612	. . . . 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 587	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ On ↦ (𝐵 +o 𝑥)) ↾ 𝐴):𝐴–1-1→(On ∖ 𝐵))
9	6	resmptd 5897	. . . . . 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 2792	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ On ↦ (𝐵 +o 𝑥)) ↾ 𝐴) = 𝐹)
12		f1eq1 6599	. . . . 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 17	. . . 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 6661	. . 3 ⊢ (𝐹:𝐴–1-1→(On ∖ 𝐵) → 𝐹:𝐴–1-1-onto→ran 𝐹)
16	14, 15	syl 17	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:𝐴–1-1-onto→ran 𝐹)
17		f1f 6604	. . . 4 ⊢ (𝐹:𝐴–1-1→(On ∖ 𝐵) → 𝐹:𝐴⟶(On ∖ 𝐵))
18		frn 6541	. . . 4 ⊢ (𝐹:𝐴⟶(On ∖ 𝐵) → ran 𝐹 ⊆ (On ∖ 𝐵))
19	14, 17, 18	3syl 18	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ (On ∖ 𝐵))
20	19	difss2d 4039	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ On)
21		reldisj 4356	. . . 4 ⊢ (ran 𝐹 ⊆ On → ((ran 𝐹 ∩ 𝐵) = ∅ ↔ ran 𝐹 ⊆ (On ∖ 𝐵)))
22	20, 21	syl 17	. . 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 515	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:𝐴–1-1-onto→ran 𝐹 ∧ (ran 𝐹 ∩ 𝐵) = ∅))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110   ∖ cdif 3854   ∩ cin 3856   ⊆ wss 3857  ∅c0 4227   ↦ cmpt 5124  ran crn 5541   ↾ cres 5542  Oncon0 6202  ⟶wf 6365  –1-1→wf1 6366  –1-1-onto→wf1o 6368  (class class class)co 7202   +_o coa 8188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pr 5311  ax-un 7512

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-int 4850  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-ov 7205  df-oprab 7206  df-mpo 7207  df-om 7634  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-oadd 8195

This theorem is referenced by:  oacomf1o  8282  onadju  9790