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Theorem fmptco 5674
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 5119 . 2 Rel (𝐺𝐹)
2 funmpt 5246 . . 3 Fun (𝑥𝐴𝑇)
3 funrel 5225 . . 3 (Fun (𝑥𝐴𝑇) → Rel (𝑥𝐴𝑇))
42, 3ax-mp 5 . 2 Rel (𝑥𝐴𝑇)
5 fmptco.1 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → 𝑅𝐵)
6 eqid 2175 . . . . . . . . . . . . 13 (𝑥𝐴𝑅) = (𝑥𝐴𝑅)
75, 6fmptd 5662 . . . . . . . . . . . 12 (𝜑 → (𝑥𝐴𝑅):𝐴𝐵)
8 fmptco.2 . . . . . . . . . . . . 13 (𝜑𝐹 = (𝑥𝐴𝑅))
98feq1d 5344 . . . . . . . . . . . 12 (𝜑 → (𝐹:𝐴𝐵 ↔ (𝑥𝐴𝑅):𝐴𝐵))
107, 9mpbird 167 . . . . . . . . . . 11 (𝜑𝐹:𝐴𝐵)
11 ffun 5360 . . . . . . . . . . 11 (𝐹:𝐴𝐵 → Fun 𝐹)
1210, 11syl 14 . . . . . . . . . 10 (𝜑 → Fun 𝐹)
13 funbrfv 5546 . . . . . . . . . . 11 (Fun 𝐹 → (𝑧𝐹𝑢 → (𝐹𝑧) = 𝑢))
1413imp 124 . . . . . . . . . 10 ((Fun 𝐹𝑧𝐹𝑢) → (𝐹𝑧) = 𝑢)
1512, 14sylan 283 . . . . . . . . 9 ((𝜑𝑧𝐹𝑢) → (𝐹𝑧) = 𝑢)
1615eqcomd 2181 . . . . . . . 8 ((𝜑𝑧𝐹𝑢) → 𝑢 = (𝐹𝑧))
1716a1d 22 . . . . . . 7 ((𝜑𝑧𝐹𝑢) → (𝑢𝐺𝑤𝑢 = (𝐹𝑧)))
1817expimpd 363 . . . . . 6 (𝜑 → ((𝑧𝐹𝑢𝑢𝐺𝑤) → 𝑢 = (𝐹𝑧)))
1918pm4.71rd 394 . . . . 5 (𝜑 → ((𝑧𝐹𝑢𝑢𝐺𝑤) ↔ (𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤))))
2019exbidv 1823 . . . 4 (𝜑 → (∃𝑢(𝑧𝐹𝑢𝑢𝐺𝑤) ↔ ∃𝑢(𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤))))
21 exsimpl 1615 . . . . . . 7 (∃𝑢(𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤)) → ∃𝑢 𝑢 = (𝐹𝑧))
22 isset 2741 . . . . . . 7 ((𝐹𝑧) ∈ V ↔ ∃𝑢 𝑢 = (𝐹𝑧))
2321, 22sylibr 134 . . . . . 6 (∃𝑢(𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤)) → (𝐹𝑧) ∈ V)
2423a1i 9 . . . . 5 (𝜑 → (∃𝑢(𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤)) → (𝐹𝑧) ∈ V))
2512adantr 276 . . . . . . . 8 ((𝜑𝑧𝐴) → Fun 𝐹)
26 fdm 5363 . . . . . . . . . . 11 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
2710, 26syl 14 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝐴)
2827eleq2d 2245 . . . . . . . . 9 (𝜑 → (𝑧 ∈ dom 𝐹𝑧𝐴))
2928biimpar 297 . . . . . . . 8 ((𝜑𝑧𝐴) → 𝑧 ∈ dom 𝐹)
30 funfvex 5524 . . . . . . . 8 ((Fun 𝐹𝑧 ∈ dom 𝐹) → (𝐹𝑧) ∈ V)
3125, 29, 30syl2anc 411 . . . . . . 7 ((𝜑𝑧𝐴) → (𝐹𝑧) ∈ V)
3231adantrr 479 . . . . . 6 ((𝜑 ∧ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)) → (𝐹𝑧) ∈ V)
3332ex 115 . . . . 5 (𝜑 → ((𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇) → (𝐹𝑧) ∈ V))
34 breq2 4002 . . . . . . . . 9 (𝑢 = (𝐹𝑧) → (𝑧𝐹𝑢𝑧𝐹(𝐹𝑧)))
35 breq1 4001 . . . . . . . . 9 (𝑢 = (𝐹𝑧) → (𝑢𝐺𝑤 ↔ (𝐹𝑧)𝐺𝑤))
3634, 35anbi12d 473 . . . . . . . 8 (𝑢 = (𝐹𝑧) → ((𝑧𝐹𝑢𝑢𝐺𝑤) ↔ (𝑧𝐹(𝐹𝑧) ∧ (𝐹𝑧)𝐺𝑤)))
3736ceqsexgv 2864 . . . . . . 7 ((𝐹𝑧) ∈ V → (∃𝑢(𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤)) ↔ (𝑧𝐹(𝐹𝑧) ∧ (𝐹𝑧)𝐺𝑤)))
38 funfvbrb 5621 . . . . . . . . . . 11 (Fun 𝐹 → (𝑧 ∈ dom 𝐹𝑧𝐹(𝐹𝑧)))
3912, 38syl 14 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ dom 𝐹𝑧𝐹(𝐹𝑧)))
4039, 28bitr3d 190 . . . . . . . . 9 (𝜑 → (𝑧𝐹(𝐹𝑧) ↔ 𝑧𝐴))
418fveq1d 5509 . . . . . . . . . 10 (𝜑 → (𝐹𝑧) = ((𝑥𝐴𝑅)‘𝑧))
42 fmptco.3 . . . . . . . . . 10 (𝜑𝐺 = (𝑦𝐵𝑆))
43 eqidd 2176 . . . . . . . . . 10 (𝜑𝑤 = 𝑤)
4441, 42, 43breq123d 4012 . . . . . . . . 9 (𝜑 → ((𝐹𝑧)𝐺𝑤 ↔ ((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤))
4540, 44anbi12d 473 . . . . . . . 8 (𝜑 → ((𝑧𝐹(𝐹𝑧) ∧ (𝐹𝑧)𝐺𝑤) ↔ (𝑧𝐴 ∧ ((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤)))
46 nfcv 2317 . . . . . . . . . . 11 𝑥𝑧
47 nfv 1526 . . . . . . . . . . . 12 𝑥𝜑
48 nffvmpt1 5518 . . . . . . . . . . . . . 14 𝑥((𝑥𝐴𝑅)‘𝑧)
49 nfcv 2317 . . . . . . . . . . . . . 14 𝑥(𝑦𝐵𝑆)
50 nfcv 2317 . . . . . . . . . . . . . 14 𝑥𝑤
5148, 49, 50nfbr 4044 . . . . . . . . . . . . 13 𝑥((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤
52 nfcsb1v 3088 . . . . . . . . . . . . . 14 𝑥𝑧 / 𝑥𝑇
5352nfeq2 2329 . . . . . . . . . . . . 13 𝑥 𝑤 = 𝑧 / 𝑥𝑇
5451, 53nfbi 1587 . . . . . . . . . . . 12 𝑥(((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇)
5547, 54nfim 1570 . . . . . . . . . . 11 𝑥(𝜑 → (((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇))
56 fveq2 5507 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ((𝑥𝐴𝑅)‘𝑥) = ((𝑥𝐴𝑅)‘𝑧))
5756breq1d 4008 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤 ↔ ((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤))
58 csbeq1a 3064 . . . . . . . . . . . . . 14 (𝑥 = 𝑧𝑇 = 𝑧 / 𝑥𝑇)
5958eqeq2d 2187 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑤 = 𝑇𝑤 = 𝑧 / 𝑥𝑇))
6057, 59bibi12d 235 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤𝑤 = 𝑇) ↔ (((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇)))
6160imbi2d 230 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝜑 → (((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤𝑤 = 𝑇)) ↔ (𝜑 → (((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇))))
62 vex 2738 . . . . . . . . . . . . . 14 𝑤 ∈ V
63 simpl 109 . . . . . . . . . . . . . . . . 17 ((𝑦 = 𝑅𝑢 = 𝑤) → 𝑦 = 𝑅)
6463eleq1d 2244 . . . . . . . . . . . . . . . 16 ((𝑦 = 𝑅𝑢 = 𝑤) → (𝑦𝐵𝑅𝐵))
65 simpr 110 . . . . . . . . . . . . . . . . 17 ((𝑦 = 𝑅𝑢 = 𝑤) → 𝑢 = 𝑤)
66 fmptco.4 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑅𝑆 = 𝑇)
6766adantr 276 . . . . . . . . . . . . . . . . 17 ((𝑦 = 𝑅𝑢 = 𝑤) → 𝑆 = 𝑇)
6865, 67eqeq12d 2190 . . . . . . . . . . . . . . . 16 ((𝑦 = 𝑅𝑢 = 𝑤) → (𝑢 = 𝑆𝑤 = 𝑇))
6964, 68anbi12d 473 . . . . . . . . . . . . . . 15 ((𝑦 = 𝑅𝑢 = 𝑤) → ((𝑦𝐵𝑢 = 𝑆) ↔ (𝑅𝐵𝑤 = 𝑇)))
70 df-mpt 4061 . . . . . . . . . . . . . . 15 (𝑦𝐵𝑆) = {⟨𝑦, 𝑢⟩ ∣ (𝑦𝐵𝑢 = 𝑆)}
7169, 70brabga 4258 . . . . . . . . . . . . . 14 ((𝑅𝐵𝑤 ∈ V) → (𝑅(𝑦𝐵𝑆)𝑤 ↔ (𝑅𝐵𝑤 = 𝑇)))
725, 62, 71sylancl 413 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → (𝑅(𝑦𝐵𝑆)𝑤 ↔ (𝑅𝐵𝑤 = 𝑇)))
73 simpr 110 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴) → 𝑥𝐴)
746fvmpt2 5591 . . . . . . . . . . . . . . 15 ((𝑥𝐴𝑅𝐵) → ((𝑥𝐴𝑅)‘𝑥) = 𝑅)
7573, 5, 74syl2anc 411 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → ((𝑥𝐴𝑅)‘𝑥) = 𝑅)
7675breq1d 4008 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → (((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤𝑅(𝑦𝐵𝑆)𝑤))
775biantrurd 305 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → (𝑤 = 𝑇 ↔ (𝑅𝐵𝑤 = 𝑇)))
7872, 76, 773bitr4d 220 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤𝑤 = 𝑇))
7978expcom 116 . . . . . . . . . . 11 (𝑥𝐴 → (𝜑 → (((𝑥𝐴𝑅)‘𝑥)(𝑦𝐵𝑆)𝑤𝑤 = 𝑇)))
8046, 55, 61, 79vtoclgaf 2800 . . . . . . . . . 10 (𝑧𝐴 → (𝜑 → (((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇)))
8180impcom 125 . . . . . . . . 9 ((𝜑𝑧𝐴) → (((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤𝑤 = 𝑧 / 𝑥𝑇))
8281pm5.32da 452 . . . . . . . 8 (𝜑 → ((𝑧𝐴 ∧ ((𝑥𝐴𝑅)‘𝑧)(𝑦𝐵𝑆)𝑤) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)))
8345, 82bitrd 188 . . . . . . 7 (𝜑 → ((𝑧𝐹(𝐹𝑧) ∧ (𝐹𝑧)𝐺𝑤) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)))
8437, 83sylan9bbr 463 . . . . . 6 ((𝜑 ∧ (𝐹𝑧) ∈ V) → (∃𝑢(𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤)) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)))
8584ex 115 . . . . 5 (𝜑 → ((𝐹𝑧) ∈ V → (∃𝑢(𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤)) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇))))
8624, 33, 85pm5.21ndd 705 . . . 4 (𝜑 → (∃𝑢(𝑢 = (𝐹𝑧) ∧ (𝑧𝐹𝑢𝑢𝐺𝑤)) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)))
8720, 86bitrd 188 . . 3 (𝜑 → (∃𝑢(𝑧𝐹𝑢𝑢𝐺𝑤) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)))
88 vex 2738 . . . 4 𝑧 ∈ V
8988, 62opelco 4792 . . 3 (⟨𝑧, 𝑤⟩ ∈ (𝐺𝐹) ↔ ∃𝑢(𝑧𝐹𝑢𝑢𝐺𝑤))
90 df-mpt 4061 . . . . 5 (𝑥𝐴𝑇) = {⟨𝑥, 𝑣⟩ ∣ (𝑥𝐴𝑣 = 𝑇)}
9190eleq2i 2242 . . . 4 (⟨𝑧, 𝑤⟩ ∈ (𝑥𝐴𝑇) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑣⟩ ∣ (𝑥𝐴𝑣 = 𝑇)})
92 nfv 1526 . . . . . 6 𝑥 𝑧𝐴
9352nfeq2 2329 . . . . . 6 𝑥 𝑣 = 𝑧 / 𝑥𝑇
9492, 93nfan 1563 . . . . 5 𝑥(𝑧𝐴𝑣 = 𝑧 / 𝑥𝑇)
95 nfv 1526 . . . . 5 𝑣(𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)
96 eleq1 2238 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
9758eqeq2d 2187 . . . . . 6 (𝑥 = 𝑧 → (𝑣 = 𝑇𝑣 = 𝑧 / 𝑥𝑇))
9896, 97anbi12d 473 . . . . 5 (𝑥 = 𝑧 → ((𝑥𝐴𝑣 = 𝑇) ↔ (𝑧𝐴𝑣 = 𝑧 / 𝑥𝑇)))
99 eqeq1 2182 . . . . . 6 (𝑣 = 𝑤 → (𝑣 = 𝑧 / 𝑥𝑇𝑤 = 𝑧 / 𝑥𝑇))
10099anbi2d 464 . . . . 5 (𝑣 = 𝑤 → ((𝑧𝐴𝑣 = 𝑧 / 𝑥𝑇) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇)))
10194, 95, 88, 62, 98, 100opelopabf 4268 . . . 4 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑣⟩ ∣ (𝑥𝐴𝑣 = 𝑇)} ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇))
10291, 101bitri 184 . . 3 (⟨𝑧, 𝑤⟩ ∈ (𝑥𝐴𝑇) ↔ (𝑧𝐴𝑤 = 𝑧 / 𝑥𝑇))
10387, 89, 1023bitr4g 223 . 2 (𝜑 → (⟨𝑧, 𝑤⟩ ∈ (𝐺𝐹) ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑥𝐴𝑇)))
1041, 4, 103eqrelrdv 4716 1 (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑇))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wex 1490  wcel 2146  Vcvv 2735  csb 3055  cop 3592   class class class wbr 3998  {copab 4058  cmpt 4059  dom cdm 4620  ccom 4624  Rel wrel 4625  Fun wfun 5202  wf 5204  cfv 5208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-fv 5216
This theorem is referenced by:  fmptcof  5675  cofmpt  5677  fcompt  5678  fcoconst  5679  ofco  6091  lmcn2  13351  cdivcncfap  13658  negfcncf  13660  dvcj  13744  dvfre  13745  dvmptcjx  13757
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