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Theorem oacomf1olem 8173
 Description: Lemma for oacomf1o 8174. (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 8172 . . . . . . 7 (𝐵 ∈ On → (𝑥 ∈ On ↦ (𝐵 +o 𝑥)):On–1-1-onto→(On ∖ 𝐵))
21adantl 485 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ On ↦ (𝐵 +o 𝑥)):On–1-1-onto→(On ∖ 𝐵))
3 f1of1 6589 . . . . . 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 7485 . . . . . 6 (𝐴 ∈ On → 𝐴 ⊆ On)
65adantr 484 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ On)
7 f1ssres 6557 . . . . 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 587 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ On ↦ (𝐵 +o 𝑥)) ↾ 𝐴):𝐴1-1→(On ∖ 𝐵))
96resmptd 5875 . . . . . 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 2851 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ On ↦ (𝐵 +o 𝑥)) ↾ 𝐴) = 𝐹)
12 f1eq1 6544 . . . . 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 235 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:𝐴1-1→(On ∖ 𝐵))
15 f1f1orn 6601 . . 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 6549 . . . 4 (𝐹:𝐴1-1→(On ∖ 𝐵) → 𝐹:𝐴⟶(On ∖ 𝐵))
18 frn 6493 . . . 4 (𝐹:𝐴⟶(On ∖ 𝐵) → ran 𝐹 ⊆ (On ∖ 𝐵))
1914, 17, 183syl 18 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ (On ∖ 𝐵))
2019difss2d 4062 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ On)
21 reldisj 4359 . . . 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 260 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran 𝐹𝐵) = ∅)
2416, 23jca 515 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:𝐴1-1-onto→ran 𝐹 ∧ (ran 𝐹𝐵) = ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∖ cdif 3878   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243   ↦ cmpt 5110  ran crn 5520   ↾ cres 5521  Oncon0 6159  ⟶wf 6320  –1-1→wf1 6321  –1-1-onto→wf1o 6323  (class class class)co 7135   +o coa 8082 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-oadd 8089 This theorem is referenced by:  oacomf1o  8174  onadju  9604
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