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Theorem fpwwe2lem6 9336
 Description: Lemma for fpwwe2 9344. (Contributed by Mario Carneiro, 18-May-2015.)
Hypotheses
Ref Expression
fpwwe2.1 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
fpwwe2.2 (𝜑𝐴 ∈ V)
fpwwe2.3 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
fpwwe2lem9.x (𝜑𝑋𝑊𝑅)
fpwwe2lem9.y (𝜑𝑌𝑊𝑆)
fpwwe2lem9.m 𝑀 = OrdIso(𝑅, 𝑋)
fpwwe2lem9.n 𝑁 = OrdIso(𝑆, 𝑌)
fpwwe2lem7.1 (𝜑𝐵 ∈ dom 𝑀)
fpwwe2lem7.2 (𝜑𝐵 ∈ dom 𝑁)
fpwwe2lem7.3 (𝜑 → (𝑀𝐵) = (𝑁𝐵))
Assertion
Ref Expression
fpwwe2lem6 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝐶𝑋𝐶𝑌 ∧ (𝑀𝐶) = (𝑁𝐶)))
Distinct variable groups:   𝑦,𝑢,𝐵   𝑢,𝑟,𝑥,𝑦,𝐹   𝑋,𝑟,𝑢,𝑥,𝑦   𝑀,𝑟,𝑢,𝑥,𝑦   𝑁,𝑟,𝑢,𝑥,𝑦   𝜑,𝑟,𝑢,𝑥,𝑦   𝐴,𝑟,𝑥   𝑅,𝑟,𝑢,𝑥,𝑦   𝑌,𝑟,𝑢,𝑥,𝑦   𝑆,𝑟,𝑢,𝑥,𝑦   𝑊,𝑟,𝑢,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑦,𝑢)   𝐵(𝑥,𝑟)   𝐶(𝑥,𝑦,𝑢,𝑟)

Proof of Theorem fpwwe2lem6
StepHypRef Expression
1 fpwwe2lem9.x . . . . . . . 8 (𝜑𝑋𝑊𝑅)
2 fpwwe2.1 . . . . . . . . 9 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
3 fpwwe2.2 . . . . . . . . 9 (𝜑𝐴 ∈ V)
42, 3fpwwe2lem2 9333 . . . . . . . 8 (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
51, 4mpbid 221 . . . . . . 7 (𝜑 → ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
65simpld 474 . . . . . 6 (𝜑 → (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)))
76simprd 478 . . . . 5 (𝜑𝑅 ⊆ (𝑋 × 𝑋))
87ssbrd 4626 . . . 4 (𝜑 → (𝐶𝑅(𝑀𝐵) → 𝐶(𝑋 × 𝑋)(𝑀𝐵)))
9 brxp 5071 . . . . 5 (𝐶(𝑋 × 𝑋)(𝑀𝐵) ↔ (𝐶𝑋 ∧ (𝑀𝐵) ∈ 𝑋))
109simplbi 475 . . . 4 (𝐶(𝑋 × 𝑋)(𝑀𝐵) → 𝐶𝑋)
118, 10syl6 34 . . 3 (𝜑 → (𝐶𝑅(𝑀𝐵) → 𝐶𝑋))
1211imp 444 . 2 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝐶𝑋)
13 imassrn 5396 . . . 4 (𝑁𝐵) ⊆ ran 𝑁
14 fpwwe2lem9.y . . . . . . . . 9 (𝜑𝑌𝑊𝑆)
152relopabi 5167 . . . . . . . . . 10 Rel 𝑊
1615brrelexi 5082 . . . . . . . . 9 (𝑌𝑊𝑆𝑌 ∈ V)
1714, 16syl 17 . . . . . . . 8 (𝜑𝑌 ∈ V)
182, 3fpwwe2lem2 9333 . . . . . . . . . . 11 (𝜑 → (𝑌𝑊𝑆 ↔ ((𝑌𝐴𝑆 ⊆ (𝑌 × 𝑌)) ∧ (𝑆 We 𝑌 ∧ ∀𝑦𝑌 [(𝑆 “ {𝑦}) / 𝑢](𝑢𝐹(𝑆 ∩ (𝑢 × 𝑢))) = 𝑦))))
1914, 18mpbid 221 . . . . . . . . . 10 (𝜑 → ((𝑌𝐴𝑆 ⊆ (𝑌 × 𝑌)) ∧ (𝑆 We 𝑌 ∧ ∀𝑦𝑌 [(𝑆 “ {𝑦}) / 𝑢](𝑢𝐹(𝑆 ∩ (𝑢 × 𝑢))) = 𝑦)))
2019simprd 478 . . . . . . . . 9 (𝜑 → (𝑆 We 𝑌 ∧ ∀𝑦𝑌 [(𝑆 “ {𝑦}) / 𝑢](𝑢𝐹(𝑆 ∩ (𝑢 × 𝑢))) = 𝑦))
2120simpld 474 . . . . . . . 8 (𝜑𝑆 We 𝑌)
22 fpwwe2lem9.n . . . . . . . . 9 𝑁 = OrdIso(𝑆, 𝑌)
2322oiiso 8325 . . . . . . . 8 ((𝑌 ∈ V ∧ 𝑆 We 𝑌) → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
2417, 21, 23syl2anc 691 . . . . . . 7 (𝜑𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
2524adantr 480 . . . . . 6 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
26 isof1o 6473 . . . . . 6 (𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌) → 𝑁:dom 𝑁1-1-onto𝑌)
2725, 26syl 17 . . . . 5 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝑁:dom 𝑁1-1-onto𝑌)
28 f1ofo 6057 . . . . 5 (𝑁:dom 𝑁1-1-onto𝑌𝑁:dom 𝑁onto𝑌)
29 forn 6031 . . . . 5 (𝑁:dom 𝑁onto𝑌 → ran 𝑁 = 𝑌)
3027, 28, 293syl 18 . . . 4 ((𝜑𝐶𝑅(𝑀𝐵)) → ran 𝑁 = 𝑌)
3113, 30syl5sseq 3616 . . 3 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑁𝐵) ⊆ 𝑌)
3215brrelexi 5082 . . . . . . . . . . . . . 14 (𝑋𝑊𝑅𝑋 ∈ V)
331, 32syl 17 . . . . . . . . . . . . 13 (𝜑𝑋 ∈ V)
345simprd 478 . . . . . . . . . . . . . 14 (𝜑 → (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))
3534simpld 474 . . . . . . . . . . . . 13 (𝜑𝑅 We 𝑋)
36 fpwwe2lem9.m . . . . . . . . . . . . . 14 𝑀 = OrdIso(𝑅, 𝑋)
3736oiiso 8325 . . . . . . . . . . . . 13 ((𝑋 ∈ V ∧ 𝑅 We 𝑋) → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
3833, 35, 37syl2anc 691 . . . . . . . . . . . 12 (𝜑𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
3938adantr 480 . . . . . . . . . . 11 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
40 isof1o 6473 . . . . . . . . . . 11 (𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋) → 𝑀:dom 𝑀1-1-onto𝑋)
4139, 40syl 17 . . . . . . . . . 10 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝑀:dom 𝑀1-1-onto𝑋)
42 f1ocnvfv2 6433 . . . . . . . . . 10 ((𝑀:dom 𝑀1-1-onto𝑋𝐶𝑋) → (𝑀‘(𝑀𝐶)) = 𝐶)
4341, 12, 42syl2anc 691 . . . . . . . . 9 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑀‘(𝑀𝐶)) = 𝐶)
44 simpr 476 . . . . . . . . 9 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝐶𝑅(𝑀𝐵))
4543, 44eqbrtrd 4605 . . . . . . . 8 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑀‘(𝑀𝐶))𝑅(𝑀𝐵))
46 f1ocnv 6062 . . . . . . . . . . 11 (𝑀:dom 𝑀1-1-onto𝑋𝑀:𝑋1-1-onto→dom 𝑀)
47 f1of 6050 . . . . . . . . . . 11 (𝑀:𝑋1-1-onto→dom 𝑀𝑀:𝑋⟶dom 𝑀)
4841, 46, 473syl 18 . . . . . . . . . 10 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝑀:𝑋⟶dom 𝑀)
4948, 12ffvelrnd 6268 . . . . . . . . 9 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑀𝐶) ∈ dom 𝑀)
50 fpwwe2lem7.1 . . . . . . . . . 10 (𝜑𝐵 ∈ dom 𝑀)
5150adantr 480 . . . . . . . . 9 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝐵 ∈ dom 𝑀)
52 isorel 6476 . . . . . . . . 9 ((𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋) ∧ ((𝑀𝐶) ∈ dom 𝑀𝐵 ∈ dom 𝑀)) → ((𝑀𝐶) E 𝐵 ↔ (𝑀‘(𝑀𝐶))𝑅(𝑀𝐵)))
5339, 49, 51, 52syl12anc 1316 . . . . . . . 8 ((𝜑𝐶𝑅(𝑀𝐵)) → ((𝑀𝐶) E 𝐵 ↔ (𝑀‘(𝑀𝐶))𝑅(𝑀𝐵)))
5445, 53mpbird 246 . . . . . . 7 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑀𝐶) E 𝐵)
55 epelg 4950 . . . . . . . 8 (𝐵 ∈ dom 𝑀 → ((𝑀𝐶) E 𝐵 ↔ (𝑀𝐶) ∈ 𝐵))
5651, 55syl 17 . . . . . . 7 ((𝜑𝐶𝑅(𝑀𝐵)) → ((𝑀𝐶) E 𝐵 ↔ (𝑀𝐶) ∈ 𝐵))
5754, 56mpbid 221 . . . . . 6 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑀𝐶) ∈ 𝐵)
58 ffn 5958 . . . . . . 7 (𝑀:𝑋⟶dom 𝑀𝑀 Fn 𝑋)
59 elpreima 6245 . . . . . . 7 (𝑀 Fn 𝑋 → (𝐶 ∈ (𝑀𝐵) ↔ (𝐶𝑋 ∧ (𝑀𝐶) ∈ 𝐵)))
6048, 58, 593syl 18 . . . . . 6 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝐶 ∈ (𝑀𝐵) ↔ (𝐶𝑋 ∧ (𝑀𝐶) ∈ 𝐵)))
6112, 57, 60mpbir2and 959 . . . . 5 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝐶 ∈ (𝑀𝐵))
62 imacnvcnv 5517 . . . . 5 (𝑀𝐵) = (𝑀𝐵)
6361, 62syl6eleq 2698 . . . 4 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝐶 ∈ (𝑀𝐵))
64 fpwwe2lem7.3 . . . . . . 7 (𝜑 → (𝑀𝐵) = (𝑁𝐵))
6564adantr 480 . . . . . 6 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑀𝐵) = (𝑁𝐵))
6665rneqd 5274 . . . . 5 ((𝜑𝐶𝑅(𝑀𝐵)) → ran (𝑀𝐵) = ran (𝑁𝐵))
67 df-ima 5051 . . . . 5 (𝑀𝐵) = ran (𝑀𝐵)
68 df-ima 5051 . . . . 5 (𝑁𝐵) = ran (𝑁𝐵)
6966, 67, 683eqtr4g 2669 . . . 4 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑀𝐵) = (𝑁𝐵))
7063, 69eleqtrd 2690 . . 3 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝐶 ∈ (𝑁𝐵))
7131, 70sseldd 3569 . 2 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝐶𝑌)
7265cnveqd 5220 . . . . 5 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑀𝐵) = (𝑁𝐵))
73 dff1o3 6056 . . . . . . 7 (𝑀:dom 𝑀1-1-onto𝑋 ↔ (𝑀:dom 𝑀onto𝑋 ∧ Fun 𝑀))
7473simprbi 479 . . . . . 6 (𝑀:dom 𝑀1-1-onto𝑋 → Fun 𝑀)
75 funcnvres 5881 . . . . . 6 (Fun 𝑀(𝑀𝐵) = (𝑀 ↾ (𝑀𝐵)))
7641, 74, 753syl 18 . . . . 5 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑀𝐵) = (𝑀 ↾ (𝑀𝐵)))
77 dff1o3 6056 . . . . . . 7 (𝑁:dom 𝑁1-1-onto𝑌 ↔ (𝑁:dom 𝑁onto𝑌 ∧ Fun 𝑁))
7877simprbi 479 . . . . . 6 (𝑁:dom 𝑁1-1-onto𝑌 → Fun 𝑁)
79 funcnvres 5881 . . . . . 6 (Fun 𝑁(𝑁𝐵) = (𝑁 ↾ (𝑁𝐵)))
8027, 78, 793syl 18 . . . . 5 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑁𝐵) = (𝑁 ↾ (𝑁𝐵)))
8172, 76, 803eqtr3d 2652 . . . 4 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑀 ↾ (𝑀𝐵)) = (𝑁 ↾ (𝑁𝐵)))
8281fveq1d 6105 . . 3 ((𝜑𝐶𝑅(𝑀𝐵)) → ((𝑀 ↾ (𝑀𝐵))‘𝐶) = ((𝑁 ↾ (𝑁𝐵))‘𝐶))
83 fvres 6117 . . . 4 (𝐶 ∈ (𝑀𝐵) → ((𝑀 ↾ (𝑀𝐵))‘𝐶) = (𝑀𝐶))
8463, 83syl 17 . . 3 ((𝜑𝐶𝑅(𝑀𝐵)) → ((𝑀 ↾ (𝑀𝐵))‘𝐶) = (𝑀𝐶))
85 fvres 6117 . . . 4 (𝐶 ∈ (𝑁𝐵) → ((𝑁 ↾ (𝑁𝐵))‘𝐶) = (𝑁𝐶))
8670, 85syl 17 . . 3 ((𝜑𝐶𝑅(𝑀𝐵)) → ((𝑁 ↾ (𝑁𝐵))‘𝐶) = (𝑁𝐶))
8782, 84, 863eqtr3d 2652 . 2 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑀𝐶) = (𝑁𝐶))
8812, 71, 873jca 1235 1 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝐶𝑋𝐶𝑌 ∧ (𝑀𝐶) = (𝑁𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  [wsbc 3402   ∩ cin 3539   ⊆ wss 3540  {csn 4125   class class class wbr 4583  {copab 4642   E cep 4947   We wwe 4996   × cxp 5036  ◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804   Isom wiso 5805  (class class class)co 6549  OrdIsocoi 8297 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-wrecs 7294  df-recs 7355  df-oi 8298 This theorem is referenced by:  fpwwe2lem7  9337
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