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Theorem elmapresaunres2 37856
Description: fresaunres2 6238 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.)
Assertion
Ref Expression
elmapresaunres2 ((𝐹 ∈ (𝐶𝑚 𝐴) ∧ 𝐺 ∈ (𝐶𝑚 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐵) = 𝐺)

Proof of Theorem elmapresaunres2
StepHypRef Expression
1 elmapi 8048 . 2 (𝐹 ∈ (𝐶𝑚 𝐴) → 𝐹:𝐴𝐶)
2 elmapi 8048 . 2 (𝐺 ∈ (𝐶𝑚 𝐵) → 𝐺:𝐵𝐶)
3 id 22 . 2 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) → (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)))
4 fresaunres2 6238 . 2 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐵) = 𝐺)
51, 2, 3, 4syl3an 1164 1 ((𝐹 ∈ (𝐶𝑚 𝐴) ∧ 𝐺 ∈ (𝐶𝑚 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐵) = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1632  wcel 2140  cun 3714  cin 3715  cres 5269  wf 6046  (class class class)co 6815  𝑚 cmap 8026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-1st 7335  df-2nd 7336  df-map 8028
This theorem is referenced by:  diophin  37857  eldioph4b  37896
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