Theorem imasetpreimafvbijlemfv 43611

Description: Lemma for imasetpreimafvbij 43615: the value of the mapping 𝐻 at a preimage of a value of function 𝐹. (Contributed by AV, 5-Mar-2024.)

Hypotheses

Ref	Expression
fundcmpsurinj.p	⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
fundcmpsurinj.h	⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))

Assertion

Ref	Expression
imasetpreimafvbijlemfv	⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → (𝐻‘𝑌) = (𝐹‘𝑋))

Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧,𝑝   𝑃,𝑝   𝑋,𝑝   𝐴,𝑝,𝑥,𝑧   𝑥,𝑃   𝑥,𝑋   𝑌,𝑝,𝑥,𝑧

Allowed substitution hints:   𝑃(𝑧)   𝐻(𝑥,𝑧,𝑝)   𝑋(𝑧)

Proof of Theorem imasetpreimafvbijlemfv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnfun 6453	. . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2	1	anim1i 616	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃) → (Fun 𝐹 ∧ 𝑌 ∈ 𝑃))
3	2	3adant3 1128	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → (Fun 𝐹 ∧ 𝑌 ∈ 𝑃))
4		fundcmpsurinj.p	. . . 4 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
5		fundcmpsurinj.h	. . . 4 ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))
6	4, 5	fundcmpsurinjlem3 43609	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ 𝑃) → (𝐻‘𝑌) = ∪ (𝐹 “ 𝑌))
7	3, 6	syl 17	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → (𝐻‘𝑌) = ∪ (𝐹 “ 𝑌))
8	1	3ad2ant1 1129	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → Fun 𝐹)
9		funiunfv 7007	. . 3 ⊢ (Fun 𝐹 → ∪ 𝑦 ∈ 𝑌 (𝐹‘𝑦) = ∪ (𝐹 “ 𝑌))
10	8, 9	syl 17	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → ∪ 𝑦 ∈ 𝑌 (𝐹‘𝑦) = ∪ (𝐹 “ 𝑌))
11		simp3 1134	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → 𝑋 ∈ 𝑌)
12		simpl1 1187	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) ∧ 𝑦 ∈ 𝑌) → 𝐹 Fn 𝐴)
13		simpl2 1188	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) ∧ 𝑦 ∈ 𝑌) → 𝑌 ∈ 𝑃)
14		simpr 487	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
15		simpl3 1189	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) ∧ 𝑦 ∈ 𝑌) → 𝑋 ∈ 𝑌)
16	4	elsetpreimafveqfv 43601	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑌 ∈ 𝑃 ∧ 𝑦 ∈ 𝑌 ∧ 𝑋 ∈ 𝑌)) → (𝐹‘𝑦) = (𝐹‘𝑋))
17	12, 13, 14, 15, 16	syl13anc 1368	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) ∧ 𝑦 ∈ 𝑌) → (𝐹‘𝑦) = (𝐹‘𝑋))
18	17	ralrimiva 3182	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → ∀𝑦 ∈ 𝑌 (𝐹‘𝑦) = (𝐹‘𝑋))
19		fveq2 6670	. . . 4 ⊢ (𝑦 = 𝑋 → (𝐹‘𝑦) = (𝐹‘𝑋))
20	19	iuneqconst 4930	. . 3 ⊢ ((𝑋 ∈ 𝑌 ∧ ∀𝑦 ∈ 𝑌 (𝐹‘𝑦) = (𝐹‘𝑋)) → ∪ 𝑦 ∈ 𝑌 (𝐹‘𝑦) = (𝐹‘𝑋))
21	11, 18, 20	syl2anc 586	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → ∪ 𝑦 ∈ 𝑌 (𝐹‘𝑦) = (𝐹‘𝑋))
22	7, 10, 21	3eqtr2d 2862	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → (𝐻‘𝑌) = (𝐹‘𝑋))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  {cab 2799  ∀wral 3138  ∃wrex 3139  {csn 4567  ∪ cuni 4838  ∪ ciun 4919   ↦ cmpt 5146  ^◡ccnv 5554   “ cima 5558  Fun wfun 6349   Fn wfn 6350  ‘cfv 6355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-fv 6363

This theorem is referenced by:  imasetpreimafvbijlemfv1  43612