New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  eqfnfv GIF version

Theorem eqfnfv 5392
 Description: Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 3-Aug-1994.) (Revised by set.mm contributors, 22-Oct-2011.)
Assertion
Ref Expression
eqfnfv ((F Fn A G Fn A) → (F = Gx A (Fx) = (Gx)))
Distinct variable groups:   x,A   x,F   x,G

Proof of Theorem eqfnfv
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 fveq1 5327 . . 3 (F = G → (Fx) = (Gx))
21ralrimivw 2698 . 2 (F = Gx A (Fx) = (Gx))
3 pm2.27 35 . . . . . . . . 9 (x A → ((x A → (Fx) = (Gx)) → (Fx) = (Gx)))
43adantl 452 . . . . . . . 8 (((F Fn A G Fn A) x A) → ((x A → (Fx) = (Gx)) → (Fx) = (Gx)))
5 eqeq1 2359 . . . . . . . . 9 ((Fx) = (Gx) → ((Fx) = y ↔ (Gx) = y))
6 fnopfvb 5359 . . . . . . . . . . 11 ((F Fn A x A) → ((Fx) = yx, y F))
76adantlr 695 . . . . . . . . . 10 (((F Fn A G Fn A) x A) → ((Fx) = yx, y F))
8 fnopfvb 5359 . . . . . . . . . . 11 ((G Fn A x A) → ((Gx) = yx, y G))
98adantll 694 . . . . . . . . . 10 (((F Fn A G Fn A) x A) → ((Gx) = yx, y G))
107, 9bibi12d 312 . . . . . . . . 9 (((F Fn A G Fn A) x A) → (((Fx) = y ↔ (Gx) = y) ↔ (x, y Fx, y G)))
115, 10syl5ib 210 . . . . . . . 8 (((F Fn A G Fn A) x A) → ((Fx) = (Gx) → (x, y Fx, y G)))
124, 11syld 40 . . . . . . 7 (((F Fn A G Fn A) x A) → ((x A → (Fx) = (Gx)) → (x, y Fx, y G)))
1312expcom 424 . . . . . 6 (x A → ((F Fn A G Fn A) → ((x A → (Fx) = (Gx)) → (x, y Fx, y G))))
14 opeldm 4910 . . . . . . . . . . . . 13 (x, y Fx dom F)
15 fndm 5182 . . . . . . . . . . . . . 14 (F Fn A → dom F = A)
1615eleq2d 2420 . . . . . . . . . . . . 13 (F Fn A → (x dom Fx A))
1714, 16syl5ib 210 . . . . . . . . . . . 12 (F Fn A → (x, y Fx A))
1817adantr 451 . . . . . . . . . . 11 ((F Fn A G Fn A) → (x, y Fx A))
1918con3d 125 . . . . . . . . . 10 ((F Fn A G Fn A) → (¬ x A → ¬ x, y F))
2019impcom 419 . . . . . . . . 9 ((¬ x A (F Fn A G Fn A)) → ¬ x, y F)
21 opeldm 4910 . . . . . . . . . . . . 13 (x, y Gx dom G)
22 fndm 5182 . . . . . . . . . . . . . 14 (G Fn A → dom G = A)
2322eleq2d 2420 . . . . . . . . . . . . 13 (G Fn A → (x dom Gx A))
2421, 23syl5ib 210 . . . . . . . . . . . 12 (G Fn A → (x, y Gx A))
2524adantl 452 . . . . . . . . . . 11 ((F Fn A G Fn A) → (x, y Gx A))
2625con3d 125 . . . . . . . . . 10 ((F Fn A G Fn A) → (¬ x A → ¬ x, y G))
2726impcom 419 . . . . . . . . 9 ((¬ x A (F Fn A G Fn A)) → ¬ x, y G)
2820, 272falsed 340 . . . . . . . 8 ((¬ x A (F Fn A G Fn A)) → (x, y Fx, y G))
2928ex 423 . . . . . . 7 x A → ((F Fn A G Fn A) → (x, y Fx, y G)))
3029a1dd 42 . . . . . 6 x A → ((F Fn A G Fn A) → ((x A → (Fx) = (Gx)) → (x, y Fx, y G))))
3113, 30pm2.61i 156 . . . . 5 ((F Fn A G Fn A) → ((x A → (Fx) = (Gx)) → (x, y Fx, y G)))
3231alrimdv 1633 . . . 4 ((F Fn A G Fn A) → ((x A → (Fx) = (Gx)) → y(x, y Fx, y G)))
3332alimdv 1621 . . 3 ((F Fn A G Fn A) → (x(x A → (Fx) = (Gx)) → xy(x, y Fx, y G)))
34 df-ral 2619 . . 3 (x A (Fx) = (Gx) ↔ x(x A → (Fx) = (Gx)))
35 eqrel 4845 . . 3 (F = Gxy(x, y Fx, y G))
3633, 34, 353imtr4g 261 . 2 ((F Fn A G Fn A) → (x A (Fx) = (Gx) → F = G))
372, 36impbid2 195 1 ((F Fn A G Fn A) → (F = Gx A (Fx) = (Gx)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ⟨cop 4561  dom cdm 4772   Fn wfn 4776   ‘cfv 4781 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-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-fv 4795 This theorem is referenced by:  eqfnfv2  5393  eqfnfvd  5395  eqfnfv2f  5396  fvreseq  5398  fconst2g  5452
 Copyright terms: Public domain W3C validator