Theorem oacomf1olem 8499

Description: Lemma for oacomf1o 8500. (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 8498	. . . . . . 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 6779	. . . . . 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 7739	. . . . . 6 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
6	5	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ On)
7		f1ssres 6743	. . . . 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 6005	. . . . . 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 2789	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ On ↦ (𝐵 +o 𝑥)) ↾ 𝐴) = 𝐹)
12		f1eq1 6731	. . . . 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 6791	. . 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 6736	. . . 4 ⊢ (𝐹:𝐴–1-1→(On ∖ 𝐵) → 𝐹:𝐴⟶(On ∖ 𝐵))
18		frn 6675	. . . 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 4079	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ On)
21		reldisj 4393	. . . 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 3886   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273   ↦ cmpt 5166  ran crn 5632   ↾ cres 5633  Oncon0 6323  ⟶wf 6494  –1-1→wf1 6495  –1-1-onto→wf1o 6497  (class class class)co 7367   +_o coa 8402

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-oadd 8409

This theorem is referenced by:  oacomf1o  8500  onadju  10116