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Theorem fmptco 5357
Description: Composition of two functions expressed as ordered-pair class abstractions. If 𝐹 has the equation ( x + 2 ) and 𝐺 the equation ( 3 * z ) then (𝐺𝐹) has the equation ( 3 * ( x + 2 ) ) . (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
fmptco.1 ((𝜑𝑥𝐴) → 𝑅𝐵)
fmptco.2 (𝜑𝐹 = (𝑥𝐴𝑅))
fmptco.3 (𝜑𝐺 = (𝑦𝐵𝑆))
fmptco.4 (𝑦 = 𝑅𝑆 = 𝑇)
Assertion
Ref Expression
fmptco (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑇))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵   𝑦,𝑅   𝜑,𝑥   𝑥,𝑆   𝑦,𝑇
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝑅(𝑥)   𝑆(𝑦)   𝑇(𝑥)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem fmptco
Dummy variables 𝑣 𝑢 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relco 4846 . 2 Rel (𝐺𝐹)
2 funmpt 4965 . . 3 Fun (𝑥𝐴𝑇)
3 funrel 4946 . . 3 (Fun (𝑥𝐴𝑇) → Rel (𝑥𝐴𝑇))
42, 3ax-mp 7 . 2 Rel (𝑥𝐴𝑇)
5 fmptco.1 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → 𝑅𝐵)
6 eqid 2056 . . . . . . . . . . . . 13 (𝑥𝐴𝑅) = (𝑥𝐴𝑅)
75, 6fmptd 5349 . . . . . . . . . . . 12 (𝜑 → (𝑥𝐴𝑅):𝐴𝐵)
8 fmptco.2 . . . . . . . . . . . . 13 (𝜑𝐹 = (𝑥𝐴𝑅))
98feq1d 5061 . . . . . . . . . . . 12 (𝜑 → (𝐹:𝐴𝐵 ↔ (𝑥𝐴𝑅):𝐴𝐵))
107, 9mpbird 160 . . . . . . . . . . 11 (𝜑𝐹:𝐴𝐵)
11 ffun 5075 . . . . . . . . . . 11 (𝐹:𝐴𝐵 → Fun 𝐹)
1210, 11syl 14 . . . . . . . . . 10 (𝜑 → Fun 𝐹)
13 funbrfv 5239 . . . . . . . . . . 11 (Fun 𝐹 → (𝑧𝐹𝑢 → (𝐹𝑧) = 𝑢))
1413imp 119 . . . . . . . . . 10 ((Fun 𝐹𝑧𝐹𝑢) → (𝐹𝑧) = 𝑢)
1512, 14sylan 271 . . . . . . . . 9 ((𝜑𝑧𝐹𝑢) → (𝐹𝑧) = 𝑢)
1615eqcomd 2061 . . . . . . . 8 ((𝜑𝑧𝐹𝑢) → 𝑢 = (𝐹𝑧))
1716a1d 22 . . . . . . 7 ((𝜑𝑧𝐹𝑢) → (𝑢𝐺𝑤𝑢 = (𝐹𝑧)))
1817expimpd 349 . . . . . 6 (𝜑 → ((𝑧𝐹𝑢𝑢𝐺𝑤) → 𝑢 = (𝐹𝑧)))
1918pm4.71rd 380 . . . . 5 (𝜑 → ((𝑧𝐹𝑢𝑢𝐺𝑤) ↔ (𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤))))
2019exbidv 1722 . . . 4 (𝜑 → (∃𝑢(𝑧𝐹𝑢𝑢𝐺𝑤) ↔ ∃𝑢(𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤))))
21 exsimpl 1524 . . . . . . 7 (∃𝑢(𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤)) → ∃𝑢 𝑢 = (𝐹𝑧))
22 isset 2578 . . . . . . 7 ((𝐹𝑧) ∈ V ↔ ∃𝑢 𝑢 = (𝐹𝑧))
2321, 22sylibr 141 . . . . . 6 (∃𝑢(𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤)) → (𝐹𝑧) ∈ V)
2423a1i 9 . . . . 5 (𝜑 → (∃𝑢(𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤)) → (𝐹𝑧) ∈ V))
2512adantr 265 . . . . . . . 8 ((𝜑𝑧𝐴) → Fun 𝐹)
26 fdm 5077 . . . . . . . . . . 11 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
2710, 26syl 14 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝐴)
2827eleq2d 2123 . . . . . . . . 9 (𝜑 → (𝑧 ∈ dom 𝐹𝑧𝐴))
2928biimpar 285 . . . . . . . 8 ((𝜑𝑧𝐴) → 𝑧 ∈ dom 𝐹)
30 funfvex 5219 . . . . . . . 8 ((Fun 𝐹𝑧 ∈ dom 𝐹) → (𝐹𝑧) ∈ V)
3125, 29, 30syl2anc 397 . . . . . . 7 ((𝜑𝑧𝐴) → (𝐹𝑧) ∈ V)
3231adantrr 456 . . . . . 6 ((𝜑 ∧ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)) → (𝐹𝑧) ∈ V)
3332ex 112 . . . . 5 (𝜑 → ((𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇) → (𝐹𝑧) ∈ V))
34 breq2 3795 . . . . . . . . 9 (𝑢 = (𝐹𝑧) → (𝑧𝐹𝑢𝑧𝐹(𝐹𝑧)))
35 breq1 3794 . . . . . . . . 9 (𝑢 = (𝐹𝑧) → (𝑢𝐺𝑤 ↔ (𝐹𝑧)𝐺𝑤))
3634, 35anbi12d 450 . . . . . . . 8 (𝑢 = (𝐹𝑧) → ((𝑧𝐹𝑢𝑢𝐺𝑤) ↔ (𝑧𝐹(𝐹𝑧) ∧ (𝐹𝑧)𝐺𝑤)))
3736ceqsexgv 2695 . . . . . . 7 ((𝐹𝑧) ∈ V → (∃𝑢(𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤)) ↔ (𝑧𝐹(𝐹𝑧) ∧ (𝐹𝑧)𝐺𝑤)))
38 funfvbrb 5307 . . . . . . . . . . 11 (Fun 𝐹 → (𝑧 ∈ dom 𝐹𝑧𝐹(𝐹𝑧)))
3912, 38syl 14 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ dom 𝐹𝑧𝐹(𝐹𝑧)))
4039, 28bitr3d 183 . . . . . . . . 9 (𝜑 → (𝑧𝐹(𝐹𝑧) ↔ 𝑧𝐴))
418fveq1d 5207 . . . . . . . . . 10 (𝜑 → (𝐹𝑧) = ((𝑥𝐴𝑅)‘𝑧))
42 fmptco.3 . . . . . . . . . 10 (𝜑𝐺 = (𝑦𝐵𝑆))
43 eqidd 2057 . . . . . . . . . 10 (𝜑𝑤 = 𝑤)
4441, 42, 43breq123d 3805 . . . . . . . . 9 (𝜑 → ((𝐹𝑧)𝐺𝑤 ↔ ((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤))
4540, 44anbi12d 450 . . . . . . . 8 (𝜑 → ((𝑧𝐹(𝐹𝑧) ∧ (𝐹𝑧)𝐺𝑤) ↔ (𝑧𝐴 ∧ ((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤)))
46 nfcv 2194 . . . . . . . . . . 11 𝑥𝑧
47 nfv 1437 . . . . . . . . . . . 12 𝑥𝜑
48 nffvmpt1 5213 . . . . . . . . . . . . . 14 𝑥((𝑥𝐴𝑅)‘𝑧)
49 nfcv 2194 . . . . . . . . . . . . . 14 𝑥(𝑦𝐵𝑆)
50 nfcv 2194 . . . . . . . . . . . . . 14 𝑥𝑤
5148, 49, 50nfbr 3835 . . . . . . . . . . . . 13 𝑥((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤
52 nfcsb1v 2909 . . . . . . . . . . . . . 14 𝑥𝑧 / 𝑥𝑇
5352nfeq2 2205 . . . . . . . . . . . . 13 𝑥 𝑤 = 𝑧 / 𝑥𝑇
5451, 53nfbi 1497 . . . . . . . . . . . 12 𝑥(((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇)
5547, 54nfim 1480 . . . . . . . . . . 11 𝑥(𝜑 → (((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇))
56 fveq2 5205 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ((𝑥𝐴𝑅)‘𝑥) = ((𝑥𝐴𝑅)‘𝑧))
5756breq1d 3801 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤 ↔ ((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤))
58 csbeq1a 2887 . . . . . . . . . . . . . 14 (𝑥 = 𝑧𝑇 = 𝑧 / 𝑥𝑇)
5958eqeq2d 2067 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑤 = 𝑇𝑤 = 𝑧 / 𝑥𝑇))
6057, 59bibi12d 228 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤𝑤 = 𝑇) ↔ (((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇)))
6160imbi2d 223 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝜑 → (((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤𝑤 = 𝑇)) ↔ (𝜑 → (((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇))))
62 vex 2577 . . . . . . . . . . . . . 14 𝑤 ∈ V
63 simpl 106 . . . . . . . . . . . . . . . . 17 ((𝑦 = 𝑅𝑢 = 𝑤) → 𝑦 = 𝑅)
6463eleq1d 2122 . . . . . . . . . . . . . . . 16 ((𝑦 = 𝑅𝑢 = 𝑤) → (𝑦𝐵𝑅𝐵))
65 simpr 107 . . . . . . . . . . . . . . . . 17 ((𝑦 = 𝑅𝑢 = 𝑤) → 𝑢 = 𝑤)
66 fmptco.4 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑅𝑆 = 𝑇)
6766adantr 265 . . . . . . . . . . . . . . . . 17 ((𝑦 = 𝑅𝑢 = 𝑤) → 𝑆 = 𝑇)
6865, 67eqeq12d 2070 . . . . . . . . . . . . . . . 16 ((𝑦 = 𝑅𝑢 = 𝑤) → (𝑢 = 𝑆𝑤 = 𝑇))
6964, 68anbi12d 450 . . . . . . . . . . . . . . 15 ((𝑦 = 𝑅𝑢 = 𝑤) → ((𝑦𝐵𝑢 = 𝑆) ↔ (𝑅𝐵𝑤 = 𝑇)))
70 df-mpt 3847 . . . . . . . . . . . . . . 15 (𝑦𝐵𝑆) = {⟨𝑦, 𝑢⟩ ∣ (𝑦𝐵𝑢 = 𝑆)}
7169, 70brabga 4028 . . . . . . . . . . . . . 14 ((𝑅𝐵𝑤 ∈ V) → (𝑅(𝑦𝐵𝑆)𝑤 ↔ (𝑅𝐵𝑤 = 𝑇)))
725, 62, 71sylancl 398 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → (𝑅(𝑦𝐵𝑆)𝑤 ↔ (𝑅𝐵𝑤 = 𝑇)))
73 simpr 107 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴) → 𝑥𝐴)
746fvmpt2 5281 . . . . . . . . . . . . . . 15 ((𝑥𝐴𝑅𝐵) → ((𝑥𝐴𝑅)‘𝑥) = 𝑅)
7573, 5, 74syl2anc 397 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → ((𝑥𝐴𝑅)‘𝑥) = 𝑅)
7675breq1d 3801 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → (((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤𝑅(𝑦𝐵𝑆)𝑤))
775biantrurd 293 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → (𝑤 = 𝑇 ↔ (𝑅𝐵𝑤 = 𝑇)))
7872, 76, 773bitr4d 213 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤𝑤 = 𝑇))
7978expcom 113 . . . . . . . . . . 11 (𝑥𝐴 → (𝜑 → (((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤𝑤 = 𝑇)))
8046, 55, 61, 79vtoclgaf 2635 . . . . . . . . . 10 (𝑧𝐴 → (𝜑 → (((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇)))
8180impcom 120 . . . . . . . . 9 ((𝜑𝑧𝐴) → (((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇))
8281pm5.32da 433 . . . . . . . 8 (𝜑 → ((𝑧𝐴 ∧ ((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)))
8345, 82bitrd 181 . . . . . . 7 (𝜑 → ((𝑧𝐹(𝐹𝑧) ∧ (𝐹𝑧)𝐺𝑤) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)))
8437, 83sylan9bbr 444 . . . . . 6 ((𝜑 ∧ (𝐹𝑧) ∈ V) → (∃𝑢(𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤)) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)))
8584ex 112 . . . . 5 (𝜑 → ((𝐹𝑧) ∈ V → (∃𝑢(𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤)) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇))))
8624, 33, 85pm5.21ndd 631 . . . 4 (𝜑 → (∃𝑢(𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤)) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)))
8720, 86bitrd 181 . . 3 (𝜑 → (∃𝑢(𝑧𝐹𝑢𝑢𝐺𝑤) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)))
88 vex 2577 . . . 4 𝑧 ∈ V
8988, 62opelco 4534 . . 3 (⟨𝑧, 𝑤⟩ ∈ (𝐺𝐹) ↔ ∃𝑢(𝑧𝐹𝑢𝑢𝐺𝑤))
90 df-mpt 3847 . . . . 5 (𝑥𝐴𝑇) = {⟨𝑥, 𝑣⟩ ∣ (𝑥𝐴𝑣 = 𝑇)}
9190eleq2i 2120 . . . 4 (⟨𝑧, 𝑤⟩ ∈ (𝑥𝐴𝑇) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑣⟩ ∣ (𝑥𝐴𝑣 = 𝑇)})
92 nfv 1437 . . . . . 6 𝑥 𝑧𝐴
9352nfeq2 2205 . . . . . 6 𝑥 𝑣 = 𝑧 / 𝑥𝑇
9492, 93nfan 1473 . . . . 5 𝑥(𝑧𝐴𝑣 = 𝑧 / 𝑥𝑇)
95 nfv 1437 . . . . 5 𝑣(𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)
96 eleq1 2116 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
9758eqeq2d 2067 . . . . . 6 (𝑥 = 𝑧 → (𝑣 = 𝑇𝑣 = 𝑧 / 𝑥𝑇))
9896, 97anbi12d 450 . . . . 5 (𝑥 = 𝑧 → ((𝑥𝐴𝑣 = 𝑇) ↔ (𝑧𝐴𝑣 = 𝑧 / 𝑥𝑇)))
99 eqeq1 2062 . . . . . 6 (𝑣 = 𝑤 → (𝑣 = 𝑧 / 𝑥𝑇𝑤 = 𝑧 / 𝑥𝑇))
10099anbi2d 445 . . . . 5 (𝑣 = 𝑤 → ((𝑧𝐴𝑣 = 𝑧 / 𝑥𝑇) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)))
10194, 95, 88, 62, 98, 100opelopabf 4038 . . . 4 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑣⟩ ∣ (𝑥𝐴𝑣 = 𝑇)} ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇))
10291, 101bitri 177 . . 3 (⟨𝑧, 𝑤⟩ ∈ (𝑥𝐴𝑇) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇))
10387, 89, 1023bitr4g 216 . 2 (𝜑 → (⟨𝑧, 𝑤⟩ ∈ (𝐺𝐹) ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑥𝐴𝑇)))
1041, 4, 103eqrelrdv 4463 1 (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑇))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574  csb 2879  cop 3405   class class class wbr 3791  {copab 3844  cmpt 3845  dom cdm 4372  ccom 4376  Rel wrel 4377  Fun wfun 4923  wf 4925  cfv 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-fv 4937
This theorem is referenced by:  fmptcof  5358  fcompt  5360  fcoconst  5361  ofco  5756
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