Theorem oacomf1olem 8503

Description: Lemma for oacomf1o 8504. (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 8502	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝑥 ∈ On ↦ (𝐵 +o 𝑥)):On–1-1-onto→(On ∖ 𝐵))
2	1	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ On ↦ (𝐵 +o 𝑥)):On–1-1-onto→(On ∖ 𝐵))
3		f1of1 6783	. . . . . 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 7742	. . . . . 6 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
6	5	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ On)
7		f1ssres 6747	. . . . 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 585	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ On ↦ (𝐵 +o 𝑥)) ↾ 𝐴):𝐴–1-1→(On ∖ 𝐵))
9	6	resmptd 6009	. . . . . 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 2790	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ On ↦ (𝐵 +o 𝑥)) ↾ 𝐴) = 𝐹)
12		f1eq1 6735	. . . . 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 232	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:𝐴–1-1→(On ∖ 𝐵))
15		f1f1orn 6795	. . 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 6740	. . . 4 ⊢ (𝐹:𝐴–1-1→(On ∖ 𝐵) → 𝐹:𝐴⟶(On ∖ 𝐵))
18		frn 6679	. . . 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 4093	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ On)
21		reldisj 4407	. . . 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 257	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran 𝐹 ∩ 𝐵) = ∅)
24	16, 23	jca 511	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:𝐴–1-1-onto→ran 𝐹 ∧ (ran 𝐹 ∩ 𝐵) = ∅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287   ↦ cmpt 5181  ran crn 5635   ↾ cres 5636  Oncon0 6327  ⟶wf 6498  –1-1→wf1 6499  –1-1-onto→wf1o 6501  (class class class)co 7370   +_o coa 8406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-oadd 8413

This theorem is referenced by:  oacomf1o  8504  onadju  10118