Mathbox for Scott Fenton < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eqfunresadj Structured version   Visualization version   GIF version

 Description: Law for adjoining an element to restrictions of functions. (Contributed by Scott Fenton, 6-Dec-2021.)
Assertion
Ref Expression
eqfunresadj (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → (𝐹 ↾ (𝑋 ∪ {𝑌})) = (𝐺 ↾ (𝑋 ∪ {𝑌})))

Proof of Theorem eqfunresadj
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 5461 . 2 Rel (𝐹 ↾ (𝑋 ∪ {𝑌}))
2 relres 5461 . 2 Rel (𝐺 ↾ (𝑋 ∪ {𝑌}))
3 breq 4687 . . . . 5 ((𝐹𝑋) = (𝐺𝑋) → (𝑥(𝐹𝑋)𝑦𝑥(𝐺𝑋)𝑦))
433ad2ant2 1103 . . . 4 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → (𝑥(𝐹𝑋)𝑦𝑥(𝐺𝑋)𝑦))
5 velsn 4226 . . . . . . 7 (𝑥 ∈ {𝑌} ↔ 𝑥 = 𝑌)
6 simp33 1119 . . . . . . . . . 10 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → (𝐹𝑌) = (𝐺𝑌))
76eqeq1d 2653 . . . . . . . . 9 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → ((𝐹𝑌) = 𝑦 ↔ (𝐺𝑌) = 𝑦))
8 simp1l 1105 . . . . . . . . . 10 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → Fun 𝐹)
9 simp31 1117 . . . . . . . . . 10 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → 𝑌 ∈ dom 𝐹)
10 funbrfvb 6276 . . . . . . . . . 10 ((Fun 𝐹𝑌 ∈ dom 𝐹) → ((𝐹𝑌) = 𝑦𝑌𝐹𝑦))
118, 9, 10syl2anc 694 . . . . . . . . 9 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → ((𝐹𝑌) = 𝑦𝑌𝐹𝑦))
12 simp1r 1106 . . . . . . . . . 10 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → Fun 𝐺)
13 simp32 1118 . . . . . . . . . 10 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → 𝑌 ∈ dom 𝐺)
14 funbrfvb 6276 . . . . . . . . . 10 ((Fun 𝐺𝑌 ∈ dom 𝐺) → ((𝐺𝑌) = 𝑦𝑌𝐺𝑦))
1512, 13, 14syl2anc 694 . . . . . . . . 9 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → ((𝐺𝑌) = 𝑦𝑌𝐺𝑦))
167, 11, 153bitr3d 298 . . . . . . . 8 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → (𝑌𝐹𝑦𝑌𝐺𝑦))
17 breq1 4688 . . . . . . . . 9 (𝑥 = 𝑌 → (𝑥𝐹𝑦𝑌𝐹𝑦))
18 breq1 4688 . . . . . . . . 9 (𝑥 = 𝑌 → (𝑥𝐺𝑦𝑌𝐺𝑦))
1917, 18bibi12d 334 . . . . . . . 8 (𝑥 = 𝑌 → ((𝑥𝐹𝑦𝑥𝐺𝑦) ↔ (𝑌𝐹𝑦𝑌𝐺𝑦)))
2016, 19syl5ibrcom 237 . . . . . . 7 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → (𝑥 = 𝑌 → (𝑥𝐹𝑦𝑥𝐺𝑦)))
215, 20syl5bi 232 . . . . . 6 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → (𝑥 ∈ {𝑌} → (𝑥𝐹𝑦𝑥𝐺𝑦)))
2221pm5.32rd 673 . . . . 5 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → ((𝑥𝐹𝑦𝑥 ∈ {𝑌}) ↔ (𝑥𝐺𝑦𝑥 ∈ {𝑌})))
23 vex 3234 . . . . . 6 𝑦 ∈ V
2423brres 5437 . . . . 5 (𝑥(𝐹 ↾ {𝑌})𝑦 ↔ (𝑥𝐹𝑦𝑥 ∈ {𝑌}))
2523brres 5437 . . . . 5 (𝑥(𝐺 ↾ {𝑌})𝑦 ↔ (𝑥𝐺𝑦𝑥 ∈ {𝑌}))
2622, 24, 253bitr4g 303 . . . 4 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → (𝑥(𝐹 ↾ {𝑌})𝑦𝑥(𝐺 ↾ {𝑌})𝑦))
274, 26orbi12d 746 . . 3 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → ((𝑥(𝐹𝑋)𝑦𝑥(𝐹 ↾ {𝑌})𝑦) ↔ (𝑥(𝐺𝑋)𝑦𝑥(𝐺 ↾ {𝑌})𝑦)))
28 resundi 5445 . . . . 5 (𝐹 ↾ (𝑋 ∪ {𝑌})) = ((𝐹𝑋) ∪ (𝐹 ↾ {𝑌}))
2928breqi 4691 . . . 4 (𝑥(𝐹 ↾ (𝑋 ∪ {𝑌}))𝑦𝑥((𝐹𝑋) ∪ (𝐹 ↾ {𝑌}))𝑦)
30 brun 4736 . . . 4 (𝑥((𝐹𝑋) ∪ (𝐹 ↾ {𝑌}))𝑦 ↔ (𝑥(𝐹𝑋)𝑦𝑥(𝐹 ↾ {𝑌})𝑦))
3129, 30bitri 264 . . 3 (𝑥(𝐹 ↾ (𝑋 ∪ {𝑌}))𝑦 ↔ (𝑥(𝐹𝑋)𝑦𝑥(𝐹 ↾ {𝑌})𝑦))
32 resundi 5445 . . . . 5 (𝐺 ↾ (𝑋 ∪ {𝑌})) = ((𝐺𝑋) ∪ (𝐺 ↾ {𝑌}))
3332breqi 4691 . . . 4 (𝑥(𝐺 ↾ (𝑋 ∪ {𝑌}))𝑦𝑥((𝐺𝑋) ∪ (𝐺 ↾ {𝑌}))𝑦)
34 brun 4736 . . . 4 (𝑥((𝐺𝑋) ∪ (𝐺 ↾ {𝑌}))𝑦 ↔ (𝑥(𝐺𝑋)𝑦𝑥(𝐺 ↾ {𝑌})𝑦))
3533, 34bitri 264 . . 3 (𝑥(𝐺 ↾ (𝑋 ∪ {𝑌}))𝑦 ↔ (𝑥(𝐺𝑋)𝑦𝑥(𝐺 ↾ {𝑌})𝑦))
3627, 31, 353bitr4g 303 . 2 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → (𝑥(𝐹 ↾ (𝑋 ∪ {𝑌}))𝑦𝑥(𝐺 ↾ (𝑋 ∪ {𝑌}))𝑦))
371, 2, 36eqbrrdiv 5252 1 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → (𝐹 ↾ (𝑋 ∪ {𝑌})) = (𝐺 ↾ (𝑋 ∪ {𝑌})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ∪ cun 3605  {csn 4210   class class class wbr 4685  dom cdm 5143   ↾ cres 5145  Fun wfun 5920  ‘cfv 5926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934 This theorem is referenced by:  eqfunressuc  31786
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