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Theorem oacomf1olem 8560
Description: Lemma for oacomf1o 8561. (Contributed by Mario Carneiro, 30-May-2015.)
Hypothesis
Ref Expression
oacomf1olem.1 𝐹 = (𝑥𝐴 ↦ (𝐵 +o 𝑥))
Assertion
Ref Expression
oacomf1olem ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:𝐴1-1-onto→ran 𝐹 ∧ (ran 𝐹𝐵) = ∅))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem oacomf1olem
StepHypRef Expression
1 oaf1o 8559 . . . . . . 7 (𝐵 ∈ On → (𝑥 ∈ On ↦ (𝐵 +o 𝑥)):On–1-1-onto→(On ∖ 𝐵))
21adantl 483 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ On ↦ (𝐵 +o 𝑥)):On–1-1-onto→(On ∖ 𝐵))
3 f1of1 6829 . . . . . 6 ((𝑥 ∈ On ↦ (𝐵 +o 𝑥)):On–1-1-onto→(On ∖ 𝐵) → (𝑥 ∈ On ↦ (𝐵 +o 𝑥)):On–1-1→(On ∖ 𝐵))
42, 3syl 17 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ On ↦ (𝐵 +o 𝑥)):On–1-1→(On ∖ 𝐵))
5 onss 7767 . . . . . 6 (𝐴 ∈ On → 𝐴 ⊆ On)
65adantr 482 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ On)
7 f1ssres 6792 . . . . 5 (((𝑥 ∈ On ↦ (𝐵 +o 𝑥)):On–1-1→(On ∖ 𝐵) ∧ 𝐴 ⊆ On) → ((𝑥 ∈ On ↦ (𝐵 +o 𝑥)) ↾ 𝐴):𝐴1-1→(On ∖ 𝐵))
84, 6, 7syl2anc 585 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ On ↦ (𝐵 +o 𝑥)) ↾ 𝐴):𝐴1-1→(On ∖ 𝐵))
96resmptd 6038 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ On ↦ (𝐵 +o 𝑥)) ↾ 𝐴) = (𝑥𝐴 ↦ (𝐵 +o 𝑥)))
10 oacomf1olem.1 . . . . . 6 𝐹 = (𝑥𝐴 ↦ (𝐵 +o 𝑥))
119, 10eqtr4di 2791 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ On ↦ (𝐵 +o 𝑥)) ↾ 𝐴) = 𝐹)
12 f1eq1 6779 . . . . 5 (((𝑥 ∈ On ↦ (𝐵 +o 𝑥)) ↾ 𝐴) = 𝐹 → (((𝑥 ∈ On ↦ (𝐵 +o 𝑥)) ↾ 𝐴):𝐴1-1→(On ∖ 𝐵) ↔ 𝐹:𝐴1-1→(On ∖ 𝐵)))
1311, 12syl 17 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (((𝑥 ∈ On ↦ (𝐵 +o 𝑥)) ↾ 𝐴):𝐴1-1→(On ∖ 𝐵) ↔ 𝐹:𝐴1-1→(On ∖ 𝐵)))
148, 13mpbid 231 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:𝐴1-1→(On ∖ 𝐵))
15 f1f1orn 6841 . . 3 (𝐹:𝐴1-1→(On ∖ 𝐵) → 𝐹:𝐴1-1-onto→ran 𝐹)
1614, 15syl 17 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:𝐴1-1-onto→ran 𝐹)
17 f1f 6784 . . . 4 (𝐹:𝐴1-1→(On ∖ 𝐵) → 𝐹:𝐴⟶(On ∖ 𝐵))
18 frn 6721 . . . 4 (𝐹:𝐴⟶(On ∖ 𝐵) → ran 𝐹 ⊆ (On ∖ 𝐵))
1914, 17, 183syl 18 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ (On ∖ 𝐵))
2019difss2d 4133 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ On)
21 reldisj 4450 . . . 4 (ran 𝐹 ⊆ On → ((ran 𝐹𝐵) = ∅ ↔ ran 𝐹 ⊆ (On ∖ 𝐵)))
2220, 21syl 17 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((ran 𝐹𝐵) = ∅ ↔ ran 𝐹 ⊆ (On ∖ 𝐵)))
2319, 22mpbird 257 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran 𝐹𝐵) = ∅)
2416, 23jca 513 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:𝐴1-1-onto→ran 𝐹 ∧ (ran 𝐹𝐵) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  cdif 3944  cin 3946  wss 3947  c0 4321  cmpt 5230  ran crn 5676  cres 5677  Oncon0 6361  wf 6536  1-1wf1 6537  1-1-ontowf1o 6539  (class class class)co 7404   +o coa 8458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-oadd 8465
This theorem is referenced by:  oacomf1o  8561  onadju  10184
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