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Theorem fntxp 5804
 Description: If F and G are functions, then their tail cross product is a function over the intersection of their domains. (Contributed by SF, 24-Feb-2015.)
Assertion
Ref Expression
fntxp ((F Fn A G Fn B) → (FG) Fn (AB))

Proof of Theorem fntxp
Dummy variables a b c d x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brtxp 5783 . . . . . . . . . 10 (x(FG)yab(y = a, b xFa xGb))
2 brtxp 5783 . . . . . . . . . 10 (x(FG)zcd(z = c, d xFc xGd))
31, 2anbi12i 678 . . . . . . . . 9 ((x(FG)y x(FG)z) ↔ (ab(y = a, b xFa xGb) cd(z = c, d xFc xGd)))
4 ee4anv 1915 . . . . . . . . 9 (abcd((y = a, b xFa xGb) (z = c, d xFc xGd)) ↔ (ab(y = a, b xFa xGb) cd(z = c, d xFc xGd)))
53, 4bitr4i 243 . . . . . . . 8 ((x(FG)y x(FG)z) ↔ abcd((y = a, b xFa xGb) (z = c, d xFc xGd)))
6 an6 1261 . . . . . . . . . . 11 (((y = a, b xFa xGb) (z = c, d xFc xGd)) ↔ ((y = a, b z = c, d) (xFa xFc) (xGb xGd)))
7 fununiq 5517 . . . . . . . . . . . . . . . 16 ((Fun F xFa xFc) → a = c)
873expib 1154 . . . . . . . . . . . . . . 15 (Fun F → ((xFa xFc) → a = c))
9 fununiq 5517 . . . . . . . . . . . . . . . 16 ((Fun G xGb xGd) → b = d)
1093expib 1154 . . . . . . . . . . . . . . 15 (Fun G → ((xGb xGd) → b = d))
118, 10im2anan9 808 . . . . . . . . . . . . . 14 ((Fun F Fun G) → (((xFa xFc) (xGb xGd)) → (a = c b = d)))
12 eqeq12 2365 . . . . . . . . . . . . . . . 16 ((y = a, b z = c, d) → (y = za, b = c, d))
13 opth 4602 . . . . . . . . . . . . . . . 16 (a, b = c, d ↔ (a = c b = d))
1412, 13syl6bb 252 . . . . . . . . . . . . . . 15 ((y = a, b z = c, d) → (y = z ↔ (a = c b = d)))
1514imbi2d 307 . . . . . . . . . . . . . 14 ((y = a, b z = c, d) → ((((xFa xFc) (xGb xGd)) → y = z) ↔ (((xFa xFc) (xGb xGd)) → (a = c b = d))))
1611, 15syl5ibrcom 213 . . . . . . . . . . . . 13 ((Fun F Fun G) → ((y = a, b z = c, d) → (((xFa xFc) (xGb xGd)) → y = z)))
1716exp4a 589 . . . . . . . . . . . 12 ((Fun F Fun G) → ((y = a, b z = c, d) → ((xFa xFc) → ((xGb xGd) → y = z))))
18173impd 1165 . . . . . . . . . . 11 ((Fun F Fun G) → (((y = a, b z = c, d) (xFa xFc) (xGb xGd)) → y = z))
196, 18syl5bi 208 . . . . . . . . . 10 ((Fun F Fun G) → (((y = a, b xFa xGb) (z = c, d xFc xGd)) → y = z))
2019exlimdvv 1637 . . . . . . . . 9 ((Fun F Fun G) → (cd((y = a, b xFa xGb) (z = c, d xFc xGd)) → y = z))
2120exlimdvv 1637 . . . . . . . 8 ((Fun F Fun G) → (abcd((y = a, b xFa xGb) (z = c, d xFc xGd)) → y = z))
225, 21syl5bi 208 . . . . . . 7 ((Fun F Fun G) → ((x(FG)y x(FG)z) → y = z))
2322alrimiv 1631 . . . . . 6 ((Fun F Fun G) → z((x(FG)y x(FG)z) → y = z))
2423alrimivv 1632 . . . . 5 ((Fun F Fun G) → xyz((x(FG)y x(FG)z) → y = z))
25 dffun2 5119 . . . . 5 (Fun (FG) ↔ xyz((x(FG)y x(FG)z) → y = z))
2624, 25sylibr 203 . . . 4 ((Fun F Fun G) → Fun (FG))
27 dmtxp 5802 . . . . 5 dom (FG) = (dom F ∩ dom G)
28 ineq12 3452 . . . . 5 ((dom F = A dom G = B) → (dom F ∩ dom G) = (AB))
2927, 28syl5eq 2397 . . . 4 ((dom F = A dom G = B) → dom (FG) = (AB))
3026, 29anim12i 549 . . 3 (((Fun F Fun G) (dom F = A dom G = B)) → (Fun (FG) dom (FG) = (AB)))
3130an4s 799 . 2 (((Fun F dom F = A) (Fun G dom G = B)) → (Fun (FG) dom (FG) = (AB)))
32 df-fn 4790 . . 3 (F Fn A ↔ (Fun F dom F = A))
33 df-fn 4790 . . 3 (G Fn B ↔ (Fun G dom G = B))
3432, 33anbi12i 678 . 2 ((F Fn A G Fn B) ↔ ((Fun F dom F = A) (Fun G dom G = B)))
35 df-fn 4790 . 2 ((FG) Fn (AB) ↔ (Fun (FG) dom (FG) = (AB)))
3631, 34, 353imtr4i 257 1 ((F Fn A G Fn B) → (FG) Fn (AB))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541   = wceq 1642   ∩ cin 3208  ⟨cop 4561   class class class wbr 4639  dom cdm 4772  Fun wfun 4775   Fn wfn 4776   ⊗ ctxp 5735 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-2nd 4797  df-txp 5736 This theorem is referenced by:  xpassen  6057
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