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Theorem fundcmpsurinjimaid 44397
Description: Every function 𝐹:𝐴𝐵 can be decomposed into a surjective function onto the image (𝐹𝐴) of the domain of 𝐹 and an injective function from the image (𝐹𝐴). (Contributed by AV, 17-Mar-2024.)
Hypotheses
Ref Expression
fundcmpsurinjimaid.i 𝐼 = (𝐹𝐴)
fundcmpsurinjimaid.g 𝐺 = (𝑥𝐴 ↦ (𝐹𝑥))
fundcmpsurinjimaid.h 𝐻 = ( I ↾ 𝐼)
Assertion
Ref Expression
fundcmpsurinjimaid (𝐹:𝐴𝐵 → (𝐺:𝐴onto𝐼𝐻:𝐼1-1𝐵𝐹 = (𝐻𝐺)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝐻   𝑥,𝐼
Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem fundcmpsurinjimaid
StepHypRef Expression
1 fimadmfo 6601 . . 3 (𝐹:𝐴𝐵𝐹:𝐴onto→(𝐹𝐴))
2 fundcmpsurinjimaid.g . . . . 5 𝐺 = (𝑥𝐴 ↦ (𝐹𝑥))
3 ffn 6504 . . . . . . 7 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
4 dffn5 6728 . . . . . . 7 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
53, 4sylib 221 . . . . . 6 (𝐹:𝐴𝐵𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
65eqcomd 2744 . . . . 5 (𝐹:𝐴𝐵 → (𝑥𝐴 ↦ (𝐹𝑥)) = 𝐹)
72, 6syl5eq 2785 . . . 4 (𝐹:𝐴𝐵𝐺 = 𝐹)
8 eqidd 2739 . . . 4 (𝐹:𝐴𝐵𝐴 = 𝐴)
9 fundcmpsurinjimaid.i . . . . 5 𝐼 = (𝐹𝐴)
109a1i 11 . . . 4 (𝐹:𝐴𝐵𝐼 = (𝐹𝐴))
117, 8, 10foeq123d 6611 . . 3 (𝐹:𝐴𝐵 → (𝐺:𝐴onto𝐼𝐹:𝐴onto→(𝐹𝐴)))
121, 11mpbird 260 . 2 (𝐹:𝐴𝐵𝐺:𝐴onto𝐼)
13 f1oi 6655 . . 3 ( I ↾ 𝐼):𝐼1-1-onto𝐼
14 f1of1 6617 . . 3 (( I ↾ 𝐼):𝐼1-1-onto𝐼 → ( I ↾ 𝐼):𝐼1-1𝐼)
15 fundcmpsurinjimaid.h . . . . . . 7 𝐻 = ( I ↾ 𝐼)
16 f1eq1 6569 . . . . . . 7 (𝐻 = ( I ↾ 𝐼) → (𝐻:𝐼1-1𝐼 ↔ ( I ↾ 𝐼):𝐼1-1𝐼))
1715, 16ax-mp 5 . . . . . 6 (𝐻:𝐼1-1𝐼 ↔ ( I ↾ 𝐼):𝐼1-1𝐼)
1817biimpri 231 . . . . 5 (( I ↾ 𝐼):𝐼1-1𝐼𝐻:𝐼1-1𝐼)
19 fimass 6525 . . . . . 6 (𝐹:𝐴𝐵 → (𝐹𝐴) ⊆ 𝐵)
209, 19eqsstrid 3925 . . . . 5 (𝐹:𝐴𝐵𝐼𝐵)
21 f1ss 6580 . . . . 5 ((𝐻:𝐼1-1𝐼𝐼𝐵) → 𝐻:𝐼1-1𝐵)
2218, 20, 21syl2an 599 . . . 4 ((( I ↾ 𝐼):𝐼1-1𝐼𝐹:𝐴𝐵) → 𝐻:𝐼1-1𝐵)
2322ex 416 . . 3 (( I ↾ 𝐼):𝐼1-1𝐼 → (𝐹:𝐴𝐵𝐻:𝐼1-1𝐵))
2413, 14, 23mp2b 10 . 2 (𝐹:𝐴𝐵𝐻:𝐼1-1𝐵)
2515fveq1i 6675 . . . . 5 (𝐻‘(𝐹𝑥)) = (( I ↾ 𝐼)‘(𝐹𝑥))
263adantr 484 . . . . . . . 8 ((𝐹:𝐴𝐵𝑥𝐴) → 𝐹 Fn 𝐴)
27 simpr 488 . . . . . . . 8 ((𝐹:𝐴𝐵𝑥𝐴) → 𝑥𝐴)
2826, 27, 27fnfvimad 7007 . . . . . . 7 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ (𝐹𝐴))
2928, 9eleqtrrdi 2844 . . . . . 6 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐼)
30 fvresi 6945 . . . . . 6 ((𝐹𝑥) ∈ 𝐼 → (( I ↾ 𝐼)‘(𝐹𝑥)) = (𝐹𝑥))
3129, 30syl 17 . . . . 5 ((𝐹:𝐴𝐵𝑥𝐴) → (( I ↾ 𝐼)‘(𝐹𝑥)) = (𝐹𝑥))
3225, 31syl5eq 2785 . . . 4 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐻‘(𝐹𝑥)) = (𝐹𝑥))
3332mpteq2dva 5125 . . 3 (𝐹:𝐴𝐵 → (𝑥𝐴 ↦ (𝐻‘(𝐹𝑥))) = (𝑥𝐴 ↦ (𝐹𝑥)))
342coeq2i 5703 . . . 4 (𝐻𝐺) = (𝐻 ∘ (𝑥𝐴 ↦ (𝐹𝑥)))
35 f1of 6618 . . . . . . . 8 (( I ↾ 𝐼):𝐼1-1-onto𝐼 → ( I ↾ 𝐼):𝐼𝐼)
3613, 35ax-mp 5 . . . . . . 7 ( I ↾ 𝐼):𝐼𝐼
3715feq1i 6495 . . . . . . 7 (𝐻:𝐼𝐼 ↔ ( I ↾ 𝐼):𝐼𝐼)
3836, 37mpbir 234 . . . . . 6 𝐻:𝐼𝐼
3938a1i 11 . . . . 5 (𝐹:𝐴𝐵𝐻:𝐼𝐼)
4039, 29cofmpt 6904 . . . 4 (𝐹:𝐴𝐵 → (𝐻 ∘ (𝑥𝐴 ↦ (𝐹𝑥))) = (𝑥𝐴 ↦ (𝐻‘(𝐹𝑥))))
4134, 40syl5eq 2785 . . 3 (𝐹:𝐴𝐵 → (𝐻𝐺) = (𝑥𝐴 ↦ (𝐻‘(𝐹𝑥))))
4233, 41, 53eqtr4rd 2784 . 2 (𝐹:𝐴𝐵𝐹 = (𝐻𝐺))
4312, 24, 423jca 1129 1 (𝐹:𝐴𝐵 → (𝐺:𝐴onto𝐼𝐻:𝐼1-1𝐵𝐹 = (𝐻𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wss 3843  cmpt 5110   I cid 5428  cres 5527  cima 5528  ccom 5529   Fn wfn 6334  wf 6335  1-1wf1 6336  ontowfo 6337  1-1-ontowf1o 6338  cfv 6339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347
This theorem is referenced by:  fundcmpsurinjALT  44398
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